15-Matrices A(t) Depending on t, Derivative = dA_dt.srt

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visit MIT opencourseware at ocw.mit.edu

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所以我周末都在努力工作
so I've worked hard over the weekend

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弄清楚我上次做了什么
figured out what I was doing last time

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和我在做什么这个时间和
and what I'm doing this time and

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改进了笔记，所以你会得到一个新的
improved the notes so you'll get a new

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关于上一讲和以后的一套笔记
set of notes on the last lecture and on

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这个和我有点得到了
this one and I kind of got to got a

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我们在做什么更好的画面和
better picture of what we're doing and

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该委员会的目标是描述
that board is aiming to describe the

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我们最后做的事情的大局
large picture of what we're doing last

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时间，这次是最后一次
time and this time so last time was

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关于a时的逆变化
about changes in a inverse when a

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改变这个时间是关于变化的
changed this time is about changes in

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特征值和奇异变化
eigenvalues and changes in singular

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当你可以想象的变化时的价值观
values when a changes you can imagine

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这是一个自然重要的情况
this is a natural important situation

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矩阵移动，因此他们的
matrices move and therefore their

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反转改变了它们的特征值变化
inverses change their eigenvalues change

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他们的奇异价值会改变，而你就是这样
their singular values change and it you

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希望有一个公式，所以我们确实有一个
hope for a formula well so we did have a

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上次改变的公式
formula for last time for the change in

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反矩阵这样和我一样
the inverse matrix so that's and I

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没有得到每个U和V转置
didn't get every U and V transpose in

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视频中的正确位置，但在或
the right place in the video but or in

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笔记的第一个版本，但我
the first version of the notes but I

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希望有这样的公式
hope that there'll be the formula that

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Woodberrymorrison公式将是
that Woodberry morrison formula will be

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正确
correct

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这次确定，所以因此我不会回去了
this time ok so so I won't go back over

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那部分
that part

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但我也意识到还有另一个
but I realize also there there's another

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问题是，我们能回答的时候
question that we can answer when the

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改变时的变化非常小
change is very small when the change in

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a是da或Deltaa的一个小变化和
a is da or Delta a of a small change and

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这当然是微积分的意义所在
that's of course what calculus is about

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所以我必须要有一些平行的主题
so I have to sort of parallel topics

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这里的衍生物是什么？
here what is the derivative when the

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它的变化是无穷小的
changes of of it is infinitesimal and

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什么是实际的变化
what is the actual change when the

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变化是有限大小没关系所以现在让我
change is finite size okay so now let me

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说什么我们可以做，什么不能做
say what we can do and what we can't do

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因为哦，我会先搞清楚
for oh I'll start out by figuring out

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衍生物的反转是什么
what the derivative is for the inverse

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这就像最后一次完成一样
so that's like completing the last time

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对于无穷小的变化，我会移动
for infinitesimal changes then I'll move

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关于特征值和特征值的变化
on to changes in the eigenvalues and

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奇异的价值观，你不能
singular values and there you cannot

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期待一个精确的公式，我们有一个我们有
expect an exact formula we had a we had

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一个完全不同的公式
a formula that was exact apart from any

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这个错别字，我们会找到一个公式
typos for this and we'll find a formula

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为此，我们将找到一个公式
for this and we'll find a formula for

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这一点，对于清楚，一次会
that and for that well that one will

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来自这一个，所以这将是一个
come from this one so so this will be a

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今天突出强调特征值如何
highlight today how do the eigen values

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当矩阵改变但我们改变时
change when the matrix changes but we

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我们无法与此平行
won't be able to do parallel to this we

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我们将无法做到
won't be able to oh well we will be able

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为你做些事情的变化有限
to do something for finite changes

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这对数学来说很重要
that's important the mathematics would

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必须继续打击它的问题
have to keep hitting it that problem

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直到它到达某个地方所以我不会得到一个
until it got somewhere so I won't get an

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改变的确切公式也是如此
exact formula for the change that's too

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很多，但我会得到不平等有多大
much but I'll get inequalities how big

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这种变化可能是我能说的
that change could be what can I say

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关于它，所以这些是高度的
about it so these are that's highly

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有趣的是我可以从这开始
interesting may I start with the

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完成上一课的内容
completing the last lecture what

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逆的导数，所以我
the derivative of the inverse so I'm

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想到这里是什么设置了
thinking here so what's the setup the

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设置是我的矩阵，取决于时间
setup is my matrix a depends on time on

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T和它有一个逆逆
T and it has an inverse a inverse

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取决于T，如果我知道这一点
depends on T and if I know this

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换言之，如果我知道da
dependence in other words if I know da

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DT矩阵的含义是多少
DT how much what how the matrix is

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取决于T然后我希望我能
depending on T then I hope I could

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弄清楚a的衍生物是什么
figure out what the derivative of a

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逆是我们应该能够做到的
inverse is that we should be able to do

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这就让我开始吧
this so let me just start with the it's

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并不难，它补充了这个
not hard and it complements this one by

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做微积分的情况下将
doing the calculus case the the

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微小变化，所以我想要去
infinitesimal change so I want to get to

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我得到了我能得到的
that I'm given I'm given I can figure

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a的变化和我的工作是
out the change in a and my job is to

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找到一个改变这个a的衍生物
find a change the derivative of this a

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反之亦然所以这里很方便
and of a inverse okay so here's a handy

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身份，我只是把它放在这里
identity and I just put this here so

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这是我想要的有用身份
here's my useful identity I want to so

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就像上次我开始用一个
as last time I start with a with a

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有限变化其微积分总是如此
finite change its calculus always does

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那个权利从一个三角洲T开始
that right starts with a delta T and

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然后它变为零所以我在这里
then it goes to zero so so here I'm up

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在完整的尺寸变化，所以我认为
at the full size change so I think that

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这等于B逆a-Ba
this is equal to B inverse a minus B a

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反过来，如果它是真的那就是a
inverse and that's if it's true it's a

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非常酷的配方，看起来是真的
pretty cool formula and look it is true

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因为在这个右手边我
because over on this right hand side I

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有B反时间aa
have B inverse times a a

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与身份相比，这是我的B.
versus the identity so that's my B

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逆，我有减去B
inverse and I have the minus the B

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逆B是有一个身份
inverse B is the identity there is a

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反之，这是好的，所以这是一个
inverse it's good right so so that is a

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从那个方面我们可以真正学习
from that well we I could actually learn

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从那个等级的等级
from that the the rank of this equals

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这是一个重点
the rank of this that's that's a point

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我从大公式制造但现在
that I made from the big formula but now

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我们可以从简单的公式中看到它
we can see it from easy formula I'm

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假设B和a到处都是
assuming that B and a everywhere here

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我假设a和b是可逆的
I'm assuming that a and b are invertible

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当我乘以一个矩阵时
matrices so when i multiply by an

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可变矩阵不会改变
invertible matrix that does not change

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排名所以那些排名相同
the rank so the those have the same rank

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但我会比那更进一步
but I'm gonna get further than that I

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想找到这个没问题，那我该怎么办呢
want to find this okay so how do I go

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我该如何继续这项工作呢
how do I go forward with that job to

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很好地找到宇宙的衍生物
find the derivative of the universe well

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我要让我打电话给这个
I'm gonna let I'm gonna call this a

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在这里改变倒数
change in a inverse and over here

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我会有V会是的，好吧，看对了
I'll have V will be yeah okay see right

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是的，所以B逆将这是真的
yeah so B inverse will be this is really

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这是一个加上Delta逆转而且这个
this is a plus Delta a inverse and this

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很好，这是一个减B，所以这是
is well that's a minus B so that's

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真的减去Delta一个减去右边
really minus Delta a minus right

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变化是从A到B的变化
the change is from A to B is the change

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我正在看这个区别
here I'm looking at this the difference

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减去减B所以这是一个变化，
a minus B so it's minus a change and

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在这里，我有一个反向我没有做过
here I have a inverse I haven't done

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除了介绍这个Delta以外的任何东西
anything except to introduce this Delta

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为了让我离开它并带来
for the and get me out of it and brought

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三角洲现在好吗我想做
Delta in okay now I want to do

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计算
calculations

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所以我想B是你知道的
so I'm thinking of B as being you know a

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有点儿，有点儿
little a little there's a sort of a

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三角洲T和我双方要分
delta T and I'm gonna divide by both

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如果我，deltaTI必须这样做
sides by delta T I have to do this if I

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想要，现在我会让DeltaT归零
want and now I'll let Delta T go to zero

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所以这是最终出现的微积分
so this is calculus appears finally are

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我不会说我们的敌人微积分但是
I won't say our enemy calculus but but

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有一种类似的竞争
there's a sort of like competition

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线性代数与微积分之间的关系
between linear algebra and calculus for

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对于大学数学微积分已经有了
for college mathematics calculus has had

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这是太多的时间和注意力
far far too much time and attention it's

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喜欢它有三到四个学期
like it gets three or four semesters of

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微积分给没有得到任何人的人
calculus for people who don't get any

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线性代数我很高兴这不会出现
linear algebra I'm glad this won't be on

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视频，但我担心它无论如何
the video but I'm afraid it will anyway

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当然微积分就好了
of course calculus is fine in its place

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好吧所以现在这里的地方让Deltat
okay so here's its place now let Delta t

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归零，那么这个等式是什么
go to zero so what does this equation

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成为它刚刚结束，每个人都知道
become it just ends and everybody knows

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随着极限的ΔT去
that as the limit as delta T goes to

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零我更换三角洲的由三角洲如此
zero I replace Delta's by the Delta so

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这个Delta除以deltaT有
this Delta a divided by delta T that has

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每个顶部都有意义的意思
a meaning each the top has a meaning and

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底部有一个平均值，但随后在
the bottom has a mean but then in the

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限制它是一个有意义的比例
limit it's a ratio that has a meaning so

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da本身
da by itself

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我没有附加含义
I don't attach a meaning to that's

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无穷小这是极限所以那是
infinitesimal it's the limit so that's

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为什么我想要三角洲的delta，所以我
why I wanted a delta over a delta so I

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可以做微积分所以现在发生的事情
could do calculus so what happens now as

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deltaT变为零，当然也是
delta T goes to zero and of course as

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deltaT变为零，携带Delta
delta T goes to zero that carries Delta

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a到零，这样就变成了逆
a to zero so that becomes a inverse and

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这种方法与deltaT相比如何？
what does this approach as delta T goes

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为零
to zero

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da/dt用那个减号哦我有
da/dt with that minus sign oh I've got

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要记住减号减号是
to remember the minus sign minus sign is

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在这里-我带出了减号
in here - I'm bringing out the minus

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那么这就像我们所做的那样
sign then this was a inverse as we had

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和那个da/dt那是相反的
and that da/dt and that's a inverse

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这就是我们的公式一个很好的公式，
that's our formula a nice formula which

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某种属于人们的知识
sort of belongs in people's knowledge

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你认识到如果a是1比1
you recognize that if a was a 1 by 1

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00:11:35,770 --> 00:11:40,265
矩阵我们可以称之为X而不是a
matrix we could call it X instead of a

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00:11:40,270 --> 00:11:45,125
如果a，如果有一个1乘1的矩阵X则
if a if a there's a 1 by 1 matrix X then

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00:11:45,130 --> 00:11:46,834
我正在看到的公式
I'm seeing the formula for the

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00:11:46,839 --> 00:11:52,145
X的1的导数写一个逆
derivative of 1 over X write a inverse

185
00:11:52,150 --> 00:11:56,135
只是一个1比1的情况，它只是X超过1所以
just a 1 by 1 case it's just 1 over X so

186
00:11:56,140 --> 00:11:58,655
1或TI的衍生物应该
the derivative of 1 or maybe T I should

187
00:11:58,660 --> 00:12:02,855
如果a只是T那么说
be saying if if a is just T then the

188
00:12:02,860 --> 00:12:06,245
T相对于T的导数为1
derivative of 1 over T with respect to T

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00:12:06,250 --> 00:12:15,035
我是正方形，是正方形的负1
is is minus 1 over T squared right I've

190
00:12:15,040 --> 00:12:18,125
只是逐个案例，我们知道这是
just the one by one case we know that's

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00:12:18,130 --> 00:12:20,435
什么是微积分，现在我们能够做到
what calculus does and now we're able to

192
00:12:20,440 --> 00:12:24,454
通过n个案例来做那个就是那个
do that n by n case so that's that's

193
00:12:24,459 --> 00:12:28,985
就像好，然后是先生
just like good and then it's it sir

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00:12:28,990 --> 00:12:32,735
平行于这样的公式
parallel to formulas like this which are

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00:12:32,740 --> 00:12:37,115
这个deltaa没有变为0的地方
where this delta a is not gone to 0 it's

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00:12:37,120 --> 00:12:39,605
全尺寸但低级别
full size but low-rank

197
00:12:39,610 --> 00:12:42,065
这实际上就是公式
that was the point actually the formula

198
00:12:42,070 --> 00:12:43,685
是它会适用，如果排名
would be it would apply if the rank

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00:12:43,690 --> 00:12:46,865
虽然不低，但利息却很低
wasn't low but the interest is in low

200
00:12:46,870 --> 00:12:47,875
秩
rank

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00:12:47,880 --> 00:12:50,795
我们对此有好处吗？
are we good for this that's really the

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00:12:50,800 --> 00:12:53,695
完成最后一次讲座
completion of last times lecture with

203
00:12:53,700 --> 00:12:58,445
与衍生品确定好来回来
with derivatives ok ok come in come back

204
00:12:58,450 --> 00:13:01,925
来到这里新的东西现在lambdas
to here to the new thing now lambdas

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00:13:01,930 --> 00:13:05,345
让我们关注lambdas特征值如何
let's focus on lambdas eigenvalues how

206
00:13:05,350 --> 00:13:08,075
当特征值改变时
does the eigen value change when the

207
00:13:08,080 --> 00:13:10,985
矩阵改变请问该特征值
matrix changes how does the eigen value

208
00:13:10,990 --> 00:13:13,265
当矩阵发生变化时我会改变
change when the matrix changes so I have

209
00:13:13,270 --> 00:13:14,695
至
to

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00:13:14,700 --> 00:13:18,665
可能性一个是小变化的时候
possibilities one is small change when

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00:13:18,670 --> 00:13:20,495
我在做微积分，我让一个
I'm doing calculus and I'm letting a

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00:13:20,500 --> 00:13:25,325
deltaT变为零，另一个变满
delta T go to zero the other is full

213
00:13:25,330 --> 00:13:28,235
大小顺序改变我不会改变的地方
size order one change where I will not

214
00:13:28,240 --> 00:13:30,035
能够给你一个公式为
be able to give you a formula for the

215
00:13:30,040 --> 00:13:32,885
新的lambdas但我能告诉你
new lambdas but I'll be able to tell you

216
00:13:32,890 --> 00:13:36,305
关于他们的重要事实就是如此
important facts about them so that's so

217
00:13:36,310 --> 00:13:39,215
现在这是今天的演讲
this is this is today's lecture now you

218
00:13:39,220 --> 00:13:40,954
可以说是完成了
could say that's the completion of

219
00:13:40,959 --> 00:13:44,635
周五的演讲怎么样d拉姆达DT
Friday's lecture what about D lambda DT

220
00:13:44,640 --> 00:13:50,345
这是一个很好的公式，这是一个证据
it's a nice formula and it's a proof is

221
00:13:50,350 --> 00:13:54,204
好玩-我对这个证据感到非常高兴
fun - I was very happy about this proof

222
00:13:54,209 --> 00:13:58,024
好吧所以我猜微积分出现了
okay so I guess calculus is showing up

223
00:13:58,029 --> 00:14:00,815
在这个中间板上，我该怎么做
here on this middle board so how do I

224
00:14:00,820 --> 00:14:04,415
从特征值开始
start with the eigenvalues well start

225
00:14:04,420 --> 00:14:08,915
我所知道的，所以这些都是事实
with what I know so so these are facts

226
00:14:08,920 --> 00:14:15,785
你可以说，我必须得到
you could say that I have to get the

227
00:14:15,790 --> 00:14:17,225
特征值，当然
eigenvalues into it and of course

228
00:14:17,230 --> 00:14:19,115
特征值必须带有特征
eigenvalues have to come with eigen

229
00:14:19,120 --> 00:14:24,305
矢量，所以我会再次使用它
vectors so I'll again use a of T it'll

230
00:14:24,310 --> 00:14:27,215
取决于T和特征向量
be depending on T and an eigen vector

231
00:14:27,220 --> 00:14:30,485
这取决于T是一个特征值
that depends on T is an eigen value that

232
00:14:30,490 --> 00:14:33,155
取决于T倍的特征向量
depends on T times an eigen vector that

233
00:14:33,160 --> 00:14:38,255
取决于我们的事实
depends on T great a fact one that we

234
00:14:38,260 --> 00:14:41,635
计划采取某种方式的衍生物
plan to take the derivative of somehow

235
00:14:41,640 --> 00:14:45,065
还有第二个事实
there's also a second fact that comes

236
00:14:45,070 --> 00:14:48,785
在这里发挥什么是交易
into play here what's the deal on the

237
00:14:48,790 --> 00:14:52,985
他们就是转置的特征值
eigenvalues of a transpose they're the

238
00:14:52,990 --> 00:14:55,985
转置的特征值是相同的
same the eigenvalues of a transpose are

239
00:14:55,990 --> 00:14:57,875
与a的特征值相同
the same as the eigen values of a are

240
00:14:57,880 --> 00:15:01,535
特征向量通常不同
the eigenvectors the same not usually of

241
00:15:01,540 --> 00:15:05,584
当然，如果矩阵是对称的话
course if the matrix was symmetric then

242
00:15:05,589 --> 00:15:07,415
a和转置是一样的
a and a transpose are just the same

243
00:15:07,420 --> 00:15:09,785
所以他们会有这样的转置
thing so they would have so a transpose

244
00:15:09,790 --> 00:15:13,105
通常会有那个特征值
would have that eigenvalue but generally

245
00:15:13,110 --> 00:15:15,514
特征向量，但它通常有一个
eigenvector but it generally it has a

246
00:15:15,519 --> 00:15:20,855
不同的特征向量，它真的
different eigenvector and it really to

247
00:15:20,860 --> 00:15:22,805
保持它与这一个分开
keep it sort of separate from this one

248
00:15:22,810 --> 00:15:25,355
让我们称之为它，它会有相同的
let's call it y it'll have the same

249
00:15:25,360 --> 00:15:27,565
特征值我打算打电话
eigenvalue I'm going to call

250
00:15:27,570 --> 00:15:31,035
为什么，但我要把它变成行向量
why but I'm gonna make it a row vector

251
00:15:31,040 --> 00:15:34,885
因为转置是代替的
because a transpose is what instead of

252
00:15:34,890 --> 00:15:36,895
写下转置我会留下来的
writing down a transpose I'm gonna stay

253
00:15:36,900 --> 00:15:39,325
用a但是把特征值放在
with a but put the eigenvalue on the

254
00:15:39,330 --> 00:15:42,085
左边所以这里是特征值
left side so here here's the eigenvalue

255
00:15:42,090 --> 00:15:48,655
左边的一个特征向量，它有
eigenvector for a on the left and it has

256
00:15:48,660 --> 00:15:51,805
相同的特征值时间
the same eigenvalue times that

257
00:15:51,810 --> 00:15:54,925
特征向量，但我在向量中是一个
eigenvector but that i in vector is a

258
00:15:54,930 --> 00:15:58,555
我当然非常喜欢这个
row I am very of course this is an

259
00:15:58,560 --> 00:16:02,755
行之间的行相等于我的行
equality between rows a row times my

260
00:16:02,760 --> 00:16:05,605
矩阵给出一行，这就是
matrix gives a row so that's the

261
00:16:05,610 --> 00:16:09,325
它具有相同的特征值
eigenvalues of that it has the same

262
00:16:09,330 --> 00:16:11,755
特征值所以这是完全的
eigenvalue so this is this is totally

263
00:16:11,760 --> 00:16:14,755
平行于完全平行和
parallel to that totally parallel and

264
00:16:14,760 --> 00:16:18,295
也许有点不太确定
maybe there's sort of less definitely

265
00:16:18,300 --> 00:16:25,765
不太常见，但这只是一回事
less seen but it's just the same thing

266
00:16:25,770 --> 00:16:28,615
对于你使用每个人看到的转置
for a transpose you use everybody sees

267
00:16:28,620 --> 00:16:31,165
如果我改变这个等式，那么我就是
if I transpose this equation then I've

268
00:16:31,170 --> 00:16:33,535
得到的东西看起来像那样但是
got something that looks like that but

269
00:16:33,540 --> 00:16:36,025
我现在宁愿这样做了
I'd rather have it this way now one more

270
00:16:36,030 --> 00:16:38,305
关于我需要的这一事实的事情
thing about these one more fact I need

271
00:16:38,310 --> 00:16:42,805
我必须要有一些
there's there has to be some I some

272
00:16:42,810 --> 00:16:46,465
规范化应该怎样多久
normalization how long how what should

273
00:16:46,470 --> 00:16:48,895
是现在X可能的长度
be the length of the right now X could

274
00:16:48,900 --> 00:16:52,045
有任何长度Y可以有任何长度
have any lengths Y could have any length

275
00:16:52,050 --> 00:16:54,495
并且有一种自然的正常化
and there's a natural normalization

276
00:16:54,500 --> 00:17:00,475
这是y转置时间X等于
which is y transpose times X equal to

277
00:17:00,480 --> 00:17:06,324
规范化的一个并不能说明
one that normalizes the to doesn't tell

278
00:17:06,329 --> 00:17:08,965
我是X的长度还是Y的长度
me the length of X or the length of Y

279
00:17:08,970 --> 00:17:11,845
但它告诉我他们说的关键
but it tells me the key saying they're

280
00:17:11,850 --> 00:17:15,235
两者的长度，所以我所拥有的
lengths of both so what I've what I've

281
00:17:15,240 --> 00:17:18,715
得到那里跟踪一个本征
got there is tracking along one eigen

282
00:17:18,720 --> 00:17:22,194
值和它的一对特征向量和
value and it's pair of eigen vectors and

283
00:17:22,199 --> 00:17:24,535
你总是欢迎想到的
you're always welcome to think of the

284
00:17:24,540 --> 00:17:26,815
当y和X相同时的对称情况
symmetric case when y and X are the same

285
00:17:26,820 --> 00:17:30,115
然后我会叫他们问我哦
and then I would call them Q oh well I

286
00:17:30,120 --> 00:17:32,035
如果它是对称的，它会称它们为Q.
would call them Q if it was a symmetric

287
00:17:32,040 --> 00:17:36,325
矩阵，但如果是的话，我会的
matrix but I would have if yeah so if

288
00:17:36,330 --> 00:17:38,305
它是一个对称矩阵，我称之为
it's a symmetric matrix I would call the

289
00:17:38,310 --> 00:17:41,155
两个特征向量都称为q
both eigenvectors would be called q

290
00:17:41,160 --> 00:17:44,625
这就是说队列是一个
and this would be saying that queue is a

291
00:17:44,630 --> 00:17:48,025
单位向量，所以这是所有的东西
unit vector right so this is all stuff

292
00:17:48,030 --> 00:17:50,965
我们知道，实际上也许我应该
we know and actually maybe I should

293
00:17:50,970 --> 00:17:55,795
用矩阵表示法写出来就是这样
write it in in matrix notation just so

294
00:17:55,800 --> 00:18:05,035
因为这是重要的，如果这是一个
cuz it's important so if that's for one

295
00:18:05,040 --> 00:18:07,525
特征向量这适用于所有人
eigenvector this is for all of them at

296
00:18:07,530 --> 00:18:12,025
一旦每个人都与它的x是
once everybody's with it the x's are the

297
00:18:12,030 --> 00:18:17,095
我X的列需要仔细考虑AMDA是
columns of X and I need to mull aMDA is

298
00:18:17,100 --> 00:18:19,825
lambdas的对角矩阵和它
the diagonal matrix of lambdas and it

299
00:18:19,830 --> 00:18:21,865
必须坐在右边这样才能
has to sit on the right so that it will

300
00:18:21,870 --> 00:18:24,805
将这些列相乘，这就像
multiply those columns so this is like

301
00:18:24,810 --> 00:18:26,935
一切的一切特征向量一次
all all eigenvectors at once

302
00:18:26,940 --> 00:18:29,815
这将是什么意思
what would this one be this would be

303
00:18:29,820 --> 00:18:42,205
像Y转置等于Y转置
like Y transpose a equals a Y transpose

304
00:18:42,210 --> 00:18:46,285
Ÿs等于O和可能这些都是
Y s equals o and probably these are

305
00:18:46,290 --> 00:18:48,775
乘以它有趣我不想
multiplied it's funny I don't want to

306
00:18:48,780 --> 00:18:51,955
如果我写Y，那就说我感觉不对劲
put what I'm I feel wrong if I write Y

307
00:18:51,960 --> 00:18:56,965
像这里一样在X转置
transpose here like here the X was on

308
00:18:56,970 --> 00:18:59,245
右侧和左侧和专辑一个
the right and on the left and album one

309
00:18:59,250 --> 00:19:09,535
哦，是的，你转置是的好，所以什么
oh it yeah y transpose yeah okay so what

310
00:19:09,540 --> 00:19:17,155
我把lambdaY转置，做什么
do I put lambda Y transpose and what do

311
00:19:17,160 --> 00:19:20,005
我把这个翻译成了什么
I put here what does this translate to

312
00:19:20,010 --> 00:19:22,885
如果我这是一个特征向量
if I if this was for a one eigenvector

313
00:19:22,890 --> 00:19:25,165
对于他们所有人来说，它就是这样
for all of them at once it's just gonna

314
00:19:25,170 --> 00:19:27,745
翻译为Y转X等于
translate to Y transpose X equal the

315
00:19:27,750 --> 00:19:33,685
身份这是非常基本的东西但是
identity this is pretty basic stuff but

316
00:19:33,690 --> 00:19:35,455
不知怎的，我们并不总是
stuff somehow we don't always

317
00:19:35,460 --> 00:19:38,485
一定看还好那些是关键
necessarily see okay those are the key

318
00:19:38,490 --> 00:19:45,115
事实，现在我打算采取
facts and now I plan to take the

319
00:19:45,120 --> 00:19:47,935
衍生物带有衍生物
derivative take the derivative with

320
00:19:47,940 --> 00:19:50,965
尊重lambda哦，我可以推导出来
respect to lambda oh okay I can derive

321
00:19:50,970 --> 00:19:54,505
多了一个事实，所以这
one more fact so this

322
00:19:54,510 --> 00:20:02,215
配方这是配方一的公式之一
formula this is formula one formula one

323
00:20:02,220 --> 00:20:05,575
只是说我会得到什么，如果我打这个上
just says what do I get if I hit this on

324
00:20:05,580 --> 00:20:08,725
Y转置左边我可以做那个Y.
the left by Y transpose can I do that Y

325
00:20:08,730 --> 00:20:17,245
T等于TX的TX的转置
transpose of T a of T X of T equals

326
00:20:17,250 --> 00:20:20,605
T的lambda是我们的数字
lambda of T that's us that's a number so

327
00:20:20,610 --> 00:20:24,565
我总是可以把它带到前面
I can always bring that out but in front

328
00:20:24,570 --> 00:20:31,935
内积的矢量符号
of the inner product the vector notation

329
00:20:31,940 --> 00:20:38,414
你恳求的那样对你有好处吗？
are you good for that I'm pleading like

330
00:20:38,419 --> 00:20:43,495
我所做的一切都完全没问题
everything I've done is totally okay and

331
00:20:43,500 --> 00:20:47,695
现在我对此有所改进
now I have a improvement to make on this

332
00:20:47,700 --> 00:20:51,865
右手边哪个是y
right hand side which is so what is y

333
00:20:51,870 --> 00:20:56,155
转置时间X它是1所以让我们
transpose times X it's 1 so let's

334
00:20:56,160 --> 00:21:03,115
记住它是1，所以换句话说我
remember that it's 1 so in other words I

335
00:21:03,120 --> 00:21:06,355
有一个lambda的T的公式
have got a formula for lambda of T the

336
00:21:06,360 --> 00:21:09,955
随着时间改变矩阵的特征值
eigen value as time changes the matrix

337
00:21:09,960 --> 00:21:12,445
改变其本征值的变化
changes its eigen values change

338
00:21:12,450 --> 00:21:14,815
按照这个公式其
according to this formula its

339
00:21:14,820 --> 00:21:17,155
特征向量根据此变化
eigenvectors change according to this

340
00:21:17,160 --> 00:21:19,735
公式，它是左特征向量
formula and it's left eigenvectors

341
00:21:19,740 --> 00:21:22,794
根据那个公式改变不是
change according to that formula isn't

342
00:21:22,799 --> 00:21:24,855
所以这里的一切都是如此
that so where everything here is

343
00:21:24,860 --> 00:21:31,664
在板上，现在有什么意义
aboveboard and now what's the point

344
00:21:31,669 --> 00:21:36,775
关键是我会发现这个
the point is I'm gonna find this the

345
00:21:36,780 --> 00:21:39,325
派生所以我要接受
derivative so I'm going to take the

346
00:21:39,330 --> 00:21:42,265
该等式的衍生物，看看是什么
derivative of that equation and see what

347
00:21:42,270 --> 00:21:42,565
我明白了
I get

348
00:21:42,570 --> 00:21:45,475
这就是那个公式
that's that'll be the formula for the

349
00:21:45,480 --> 00:21:47,515
特征值的导数和它的
derivative of an eigen value and it's

350
00:21:47,520 --> 00:21:51,294
令人惊讶的是，它并不是那么广泛
it's amazingly it's not that widely

351
00:21:51,299 --> 00:21:54,405
知道它当然是经典的但是
known it's of course it's classical but

352
00:21:54,410 --> 00:21:58,495
它并不总是课程的一部分所以这些
it's not always part of courses so these

353
00:21:58,500 --> 00:22:01,554
这是随着时间变化矩阵的变化
this is as time varies the matrix varies

354
00:22:01,559 --> 00:22:04,665
a因此它的特征值
a and therefore its eigenvalues

355
00:22:04,670 --> 00:22:07,695
它是特征向量，所以我们是
and it's eigenvectors very so we're

356
00:22:07,700 --> 00:22:12,015
会发现DlambdaDT是一个级别
gonna find D lambda DT it's one level of

357
00:22:12,020 --> 00:22:16,515
更难找到DXDT那个
difficulty more to find DX DT that the

358
00:22:16,520 --> 00:22:18,975
特征向量的导数或
derivative of the eigenvector or the

359
00:22:18,980 --> 00:22:20,985
特征值的二阶导数
second derivative of the eigenvalue

360
00:22:20,990 --> 00:22:22,635
虽然那些人聚在一起
though those kind of come together and

361
00:22:22,640 --> 00:22:25,215
我不会去那里我就要去做
I'm not gonna go there I'm just gonna do

362
00:22:25,220 --> 00:22:27,705
这里有一件好事
the one great thing here take the

363
00:22:27,710 --> 00:22:30,465
我应该做那个方程的导数
derivative of that equation shall I do

364
00:22:30,470 --> 00:22:35,985
它在那边所以我们去，所以我想
it over there so here we go so I want to

365
00:22:35,990 --> 00:22:41,085
计算DlambdaDT，我正在使用它
compute D lambda DT and I'm using this

366
00:22:41,090 --> 00:22:51,135
lambda的公式，所以我有
formula for lambda there so I've got

367
00:22:51,140 --> 00:22:54,195
依赖于T和我的三件事
three things that depend on T and I'm

368
00:22:54,200 --> 00:22:56,085
采取他们的产品的衍生物
taking the derivative of their product

369
00:22:56,090 --> 00:22:59,085
所以我要正确使用产品规则
so I'm gonna use the product rule right

370
00:22:59,090 --> 00:23:01,605
现在它需要我将应用该产品
now it takes a I'll apply the product

371
00:23:01,610 --> 00:23:03,705
统治那个衍生物好吧
rule to that derivative okay

372
00:23:03,710 --> 00:23:07,775
取第一个人的衍生物
take the derivative of the first guy

373
00:23:07,780 --> 00:23:14,265
X次乘以衍生物的次数
times a times X take the derivative of

374
00:23:14,270 --> 00:23:33,695
第二个人和第二个人相遇
the second guy times the second guy and

375
00:23:33,700 --> 00:23:42,375
牛逼DX的TA的第三个家伙ÿ转
the third guy Y transpose of T a of T DX

376
00:23:42,380 --> 00:23:54,795
DT好吧我们距离一分钟一分钟
DT okay we are one minute away from a

377
00:23:54,800 --> 00:23:58,875
伟大的公式，我真的很高兴，如果
great formula and I'm really happy if

378
00:23:58,880 --> 00:24:01,695
你让我这说的是，
you allow me to say it that that the

379
00:24:01,700 --> 00:24:04,725
公式来自于采取
formula comes by just putting taking

380
00:24:04,730 --> 00:24:07,245
那些我们知道的事实
those facts we know putting them

381
00:24:07,250 --> 00:24:09,795
我们一起进入这个表达
together into this expression that we

382
00:24:09,800 --> 00:24:13,395
也知道，这就像lambda
also know and that this is like lambda

383
00:24:13,400 --> 00:24:16,325
等于X逆ax
equals X inverse ax

384
00:24:16,330 --> 00:24:19,235
这是对角化的事情然后
that's diagonalizing thing and then

385
00:24:19,240 --> 00:24:22,385
采取衍生物所以我得到了什么
taking the derivative so what do I get

386
00:24:22,390 --> 00:24:28,975
如果我这个术语能很好地利用这个衍生物
if I take that derivative well this term

387
00:24:28,980 --> 00:24:32,465
我会继续保持，好像我不会去
I'm going to keep like I'm not going to

388
00:24:32,470 --> 00:24:42,635
玩弄每个人都很清楚
play with that everybody's clear that's

389
00:24:42,640 --> 00:24:45,725
这里的数字是ADT的矩阵
a number here's a matrix the ADT is a

390
00:24:45,730 --> 00:24:49,085
矩阵我得到的导数
matrix the derivative I take the

391
00:24:49,090 --> 00:24:52,415
它的衍生物在这里的每一个条目
derivative of it every entry in a here's

392
00:24:52,420 --> 00:24:57,395
它是一个列向量，它的特征向量
it's a column vector its eigen vector

393
00:24:57,400 --> 00:24:59,615
这是Rho时间的行向量
and here's a row vector so Rho times

394
00:24:59,620 --> 00:25:02,165
矩阵时间列是第一个
matrix times column is a number one by

395
00:25:02,170 --> 00:25:06,145
一个，实际上这是我的答案
one and actually that's my answer

396
00:25:06,150 --> 00:25:08,945
那是我的回答所以我这么说
that's my answer so I'm saying that

397
00:25:08,950 --> 00:25:14,305
这两个词相互抵消了
these two terms cancel each other out

398
00:25:14,310 --> 00:25:16,925
这两个方面相加为零，并且
those two terms add to zero and that

399
00:25:16,930 --> 00:25:21,425
这是正确的答案
this is the right answer for for the

400
00:25:21,430 --> 00:25:22,355
衍生物
derivative

401
00:25:22,360 --> 00:25:26,405
这是一个很好的公式，所以找到
that's a nice formula so to find the

402
00:25:26,410 --> 00:25:30,065
您采用的特征值的导数
derivative of an eigen value you take

403
00:25:30,070 --> 00:25:33,395
矩阵改变你的速度
the rate the matrix is changing you

404
00:25:33,400 --> 00:25:36,905
乘以俄勒冈州的矢量和
multiply by the Oregon vector and by the

405
00:25:36,910 --> 00:25:39,875
左特征向量它给你一个数字
left eigen vector it gives you a number

406
00:25:39,880 --> 00:25:44,735
那就是DlambdaDT好吧，为什么呢
and that's the D lambda DT okay so why

407
00:25:44,740 --> 00:25:48,785
那两个家伙加零是不是
do those two guys add to zero that's

408
00:25:48,790 --> 00:25:50,495
那就是剩下的一切
that's all that remains here and then

409
00:25:50,500 --> 00:25:53,615
这个主题以这个好结束了
this topic is ended with this nice

410
00:25:53,620 --> 00:25:58,975
公式好，所以我想简化它
formula okay so I want to simplify that

411
00:25:58,980 --> 00:26:01,565
简化并表明他们取消了
simplify that and show that they cancel

412
00:26:01,570 --> 00:26:06,545
彼此没关系，所以什么是斧头它的lambda
each other okay so what is ax its lambda

413
00:26:06,550 --> 00:26:12,955
X所以这个家伙只不过是它的lambda
X so this guy is nothing but it's lambda

414
00:26:12,960 --> 00:26:19,625
这取决于课程时间的时间dy
that depends on time of course times dy

415
00:26:19,630 --> 00:26:23,945
DTDY转置DT我只是复制
DT D Y transpose DT I'm just copying

416
00:26:23,950 --> 00:26:27,105
那斧子拉姆达X
that ax is lambda X

417
00:26:27,110 --> 00:26:29,174
抱歉，我不是故意要看那个
sorry I didn't mean to make that look

418
00:26:29,179 --> 00:26:37,335
那个斧头很难用lambdaX和我
hard okay with that ax is lambda X and I

419
00:26:37,340 --> 00:26:39,794
我非常安全，因为lambda只是
am perfectly safe because lambda is just

420
00:26:39,799 --> 00:26:42,734
一个数字，将它带到左边
a number to bring it out to the left so

421
00:26:42,739 --> 00:26:44,624
它没有得到它看起来不像它
it's not getting doesn't look like it's

422
00:26:44,629 --> 00:26:46,784
在路上和另一个怎么样
in the way and what about this other

423
00:26:46,789 --> 00:26:54,705
长期所以我有ý转AUÿ
term so I have y transpose au y

424
00:26:54,710 --> 00:26:59,924
转置一个什么是Y.
transpose a what's that what's Y

425
00:26:59,929 --> 00:27:02,414
移调这就是组合
transpose a that's the combination that

426
00:27:02,419 --> 00:27:07,274
我知道Y转换一个让Y就是那个
I know Y transpose a a a let Y is that

427
00:27:07,279 --> 00:27:10,185
左特征向量y转置a带来
left eigenvector y transpose a is brings

428
00:27:10,190 --> 00:27:12,825
出一个lambda所以这也带来了一个
out a lambda so this also brings out a

429
00:27:12,830 --> 00:27:27,674
lambdatimesytransposexdx/dt好吧我
lambda times y transpose x dx/dt okay I

430
00:27:27,679 --> 00:27:30,674
只需使用ax等于lambdaX即可
just use ax equal lambda X there that

431
00:27:30,679 --> 00:27:33,435
现在什么都不是，我该怎么办
was really nothing now what do I do

432
00:27:33,440 --> 00:27:35,744
你能不能让我想零吗？
can you I want this to be zero can you

433
00:27:35,749 --> 00:27:38,114
看到它正在发生它是一个伟大的
see it happening it's a it's a great

434
00:27:38,119 --> 00:27:40,994
很高兴看到它发生得如此做什么
pleasure to see it happening so what do

435
00:27:40,999 --> 00:27:44,674
我在这里现在是我的第一步
I have here what's my first step now

436
00:27:44,679 --> 00:27:48,225
带来会把lambda带到那个外面
bring would bring lambda outside that's

437
00:27:48,230 --> 00:27:50,354
不是零，我们不知道那是什么
not zero we don't know what that is

438
00:27:50,359 --> 00:27:53,595
把lambda带到那里时间
bring lambda outside there times the

439
00:27:53,600 --> 00:27:59,264
因为一些奇妙的原因所以整个事情
whole thing so for some wonderful reason

440
00:27:59,269 --> 00:28:02,654
我相信这个数字是一个
I believe that this number which is a

441
00:28:02,659 --> 00:28:05,654
第一列的滚动时间乘以a
roll times at column a row times a

442
00:28:05,659 --> 00:28:09,164
列两届那里，我相信他们
column two terms there I believe they

443
00:28:09,169 --> 00:28:11,715
互相敲打他们那个
knock each other out and they that

444
00:28:11,720 --> 00:28:18,284
结果为零，为什么我为什么呢？
result is zero and why why because I

445
00:28:18,289 --> 00:28:21,705
回到这是这个董事会的
come back to the this is this board has

446
00:28:21,710 --> 00:28:25,965
所有我知道的，这就是为什么要转置
all that I know and here's why transpose

447
00:28:25,970 --> 00:28:28,335
时间X等于1，这有何帮助
times X equal one and how does that help

448
00:28:28,340 --> 00:28:31,874
我，因为我在那里看到的
me because what I'm seeing in that

449
00:28:31,879 --> 00:28:36,554
在那个方格和那些括号是
square in that and those brackets is the

450
00:28:36,559 --> 00:28:38,945
Y的导数转置X.
derivative of Y transpose X

451
00:28:38,950 --> 00:28:45,675
所以它是一个人的衍生物
so it's the derivative of one therefore

452
00:28:45,680 --> 00:28:50,445
零，所以这是它的衍生物
zero so this is the derivative of one it

453
00:28:50,450 --> 00:28:53,265
等于零这两个术语敲每个
equals zero those two terms knock each

454
00:28:53,270 --> 00:28:55,965
其他的，只留下美好的名词
other out and leave just the nice term

455
00:28:55,970 --> 00:28:58,455
我们看到的是衍生物
that we're seeing so the derivative of

456
00:28:58,460 --> 00:29:02,085
特征值只是为了有一个
the eigenvalue just just to have one

457
00:29:02,090 --> 00:29:04,785
在我们离开它之前先看看它
more look at it before we leave it the

458
00:29:04,790 --> 00:29:06,825
特征值的导数就是这个
derivative of the eigenvalue is this

459
00:29:06,830 --> 00:29:09,045
公式是它的速率
formula it's the rate at which the

460
00:29:09,050 --> 00:29:11,415
矩阵正在改变时代
matrix is changing times the

461
00:29:11,420 --> 00:29:15,555
有时左右的特征向量
eigenvectors on right and left sometimes

462
00:29:15,560 --> 00:29:17,205
他们被称为正确的特征向量
they're called the right eigenvector in

463
00:29:17,210 --> 00:29:22,655
在时间t的左特征向量
the left eigenvector at at the time t

464
00:29:22,660 --> 00:29:26,325
是的，所以我们没有看到这一点
yeah so a is so we're not seeing in this

465
00:29:26,330 --> 00:29:30,095
换言之，lambdaDT不是我
d lambda DT in other words does is not I

466
00:29:30,100 --> 00:29:34,035
不需要我得到一个很好的配方
don't need to I get a nice formula which

467
00:29:34,040 --> 00:29:36,255
不涉及的衍生物
doesn't involve the derivative of the

468
00:29:36,260 --> 00:29:39,375
特征向量，如果我是它的美丽
eigenvector that's the beauty of it if I

469
00:29:39,380 --> 00:29:42,735
想上去应该迈出下一步
want to go up should take the next step

470
00:29:42,740 --> 00:29:47,565
我本周末试过，但如果这是一团糟
I tried this weekend but it's a mess if

471
00:29:47,570 --> 00:29:49,995
这是我的意思
it would be to take there so I've this

472
00:29:50,000 --> 00:29:52,725
是我的公式和DlambdaDT等于
is my formula and D lambda DT equals

473
00:29:52,730 --> 00:29:55,815
这个和我可以采取下一个衍生物
this and I can take the next derivative

474
00:29:55,820 --> 00:29:59,115
这一点，并会涉及第二
of that and it will involve the second

475
00:29:59,120 --> 00:30:01,965
天DT正确，但也会
day DT squared right but it will also

476
00:30:01,970 --> 00:30:07,725
实际上涉及DXDT和dyDT
involve DX DT and dy DT and it in fact

477
00:30:07,730 --> 00:30:10,335
伪逆甚至显示出它
the pseudo-inverse even shows up it's

478
00:30:10,340 --> 00:30:15,915
这是另一个步骤，我不会去
it's a another step and I'm not going

479
00:30:15,920 --> 00:30:19,725
那是因为我们拥有最好的
that far because we've got the the best

480
00:30:19,730 --> 00:30:24,915
公式那里好吧所以现在有了
formula there okay so now that has

481
00:30:24,920 --> 00:30:31,065
我回答了这个问题
answered this question and I could

482
00:30:31,070 --> 00:30:33,465
以同样的方式回答这个问题
answer that question the same way it

483
00:30:33,470 --> 00:30:36,195
将涉及转置a和
would involve a transpose a and the

484
00:30:36,200 --> 00:30:39,225
奇异向量而不是涉及到
singular vectors instead of involving a

485
00:30:39,230 --> 00:30:43,445
并且特征向量可能是a
and the eigenvectors maybe that's a

486
00:30:43,450 --> 00:30:45,855
适合的运动我不知道我没有
suitable exercise I don't know I haven't

487
00:30:45,860 --> 00:30:51,445
自己做了我想做的就是这个
done it myself what I want to do is this

488
00:30:51,450 --> 00:30:55,885
现在，我们可以说什么可以说
now now it's say what can we say about

489
00:30:55,890 --> 00:30:58,835
特征值的变化，我会
the change in the eigenvalue and I'll

490
00:30:58,840 --> 00:31:01,825
首先要保留特征值
just stay first of all with eigenvalues

491
00:31:01,830 --> 00:31:07,745
当改变就像Rank1那样
when the change is like Rank 1 that way

492
00:31:07,750 --> 00:31:10,505
这是一个完美的例子
this is a perfect example when the

493
00:31:10,510 --> 00:31:18,455
变化是等级1所以所以我真的如此
change is Rank 1 so so so I'm really so

494
00:31:18,460 --> 00:31:26,255
关于特征值我们能说些什么呢？
what can we say about the eigenvalues

495
00:31:26,260 --> 00:31:28,025
让我们把最大的顶部
let's take the top the largest

496
00:31:28,030 --> 00:31:31,745
特征值或所有这些都是特征值
eigenvalue or all of them all of them

497
00:31:31,750 --> 00:31:39,995
lambdaJ他们都加了一个等级
lambda J all of them of a plus a rank

498
00:31:40,000 --> 00:31:45,185
一个矩阵UV转置让我做
one matrix UV transpose oh let me make

499
00:31:45,190 --> 00:31:50,945
它让我们，让我们在这里做一个漂亮的情况下，
it let's let's do the nice case here a

500
00:31:50,950 --> 00:31:54,395
很好的情况，因为如果我允许一般
nice case because if I allow a general

501
00:31:54,400 --> 00:31:57,395
矩阵我不得不担心它
matrix a I have to worry about does it

502
00:31:57,400 --> 00:31:59,315
有足够的特征向量你知道吗
have enough eigenvectors you know can it

503
00:31:59,320 --> 00:32:02,645
将所有那些东西对齐化
diagonalize all that stuff let's make

504
00:32:02,650 --> 00:32:06,955
让我们使它成为一个对称矩阵和
let's make it a symmetric matrix and

505
00:32:06,960 --> 00:32:11,375
让我们使等级一变化对称
let's make the rank one change symmetric

506
00:32:11,380 --> 00:32:14,525
也因此，问题是我能说什么
too so the question is what can I say

507
00:32:14,530 --> 00:32:18,875
关于排名之后的特征值
about the eigenvalues of after a rank

508
00:32:18,880 --> 00:32:22,145
一次改变，这不是微积分
one change so again this isn't calculus

509
00:32:22,150 --> 00:32:24,725
现在因为我正在做的改变
now because the change that I'm making

510
00:32:24,730 --> 00:32:28,955
完全是一个真正的向量而不是
is as fully as is a true vector and not

511
00:32:28,960 --> 00:32:33,545
差别好，我不会
a differential okay and I'm not going to

512
00:32:33,550 --> 00:32:36,755
具有用于新的固有的精确配方
have an exact formula for the new eigen

513
00:32:36,760 --> 00:32:39,845
我说的价值，但我要去做什么
values as I said but what I am going to

514
00:32:39,850 --> 00:32:46,265
做的是写下美好的事实
do is write down the beautiful facts

515
00:32:46,270 --> 00:32:49,625
那是众所周知的，在这里他们
that are known about that and here they

516
00:32:49,630 --> 00:32:54,955
首先是特征值
are so first of all the eigen values are

517
00:32:54,960 --> 00:32:59,135
按降序排列，我们使用降序
in descending order we use descending

518
00:32:59,140 --> 00:33:01,595
奇异值的顺序让我们使用它们
order for singular values let's use them

519
00:33:01,600 --> 00:33:04,715
也是因为λ为1的特征值
also for an eigen value so lambda 1

520
00:33:04,720 --> 00:33:07,105
大于等于λ2更大
is greater equal to lambda 2 greater

521
00:33:07,110 --> 00:33:10,205
等于lambda3等等
equal lambda 3 and so on

522
00:33:10,210 --> 00:33:19,835
哦，给我给我一个啊给我一个
Oh give me give me a yeah give me an

523
00:33:19,840 --> 00:33:25,414
你有什么想法？
idea well what do you expect from from

524
00:33:25,419 --> 00:33:27,695
将这一变化排在第一位
that rank one change so that that

525
00:33:27,700 --> 00:33:30,755
变化等级1你能告诉我了吗？
changes Rank 1 can you tell me any more

526
00:33:30,760 --> 00:33:33,455
关于那个改变你转移什么
about that change you you transpose what

527
00:33:33,460 --> 00:33:39,395
一种矩阵是uu转置它的
kind of a matrix is uu transpose it's

528
00:33:39,400 --> 00:33:42,355
等级1，但我们可以说更多
rank 1 but we can say more

529
00:33:42,360 --> 00:33:48,845
这是当然的负鼠埃特里克，它是
it is possum Ettrick of course and it is

530
00:33:48,850 --> 00:33:53,315
是的积极半肯定的积极
yeah positive semi definite positive

531
00:33:53,320 --> 00:33:56,095
半确定这是一个积极的变化
semi definite this is a positive change

532
00:33:56,100 --> 00:34:01,404
uutranspose是典型的排名1
uu transpose is the typical rank 1

533
00:34:01,409 --> 00:34:03,725
正半确定它不可能
positive semi definite it couldn't be

534
00:34:03,730 --> 00:34:05,284
肯定的，因为它只有
positive definite because it's only got

535
00:34:05,289 --> 00:34:08,255
等级1本征是什么的
Rank 1 what are the eigen what's the

536
00:34:08,260 --> 00:34:10,685
那个矩阵的特征向量是的，是的
eigenvector of that matrix yeah let's

537
00:34:10,690 --> 00:34:15,035
只是为什么不能在这里我们可以通过两种做到这一点
just why not here we can do this in two

538
00:34:15,040 --> 00:34:15,965
秒
seconds

539
00:34:15,970 --> 00:34:18,904
所以uutranspose这是一个矩阵我
so uu transpose that's a matrix I'm

540
00:34:18,909 --> 00:34:21,575
请您想想，这是一个
asking you to think about and it's a

541
00:34:21,580 --> 00:34:24,424
全n乘n矩阵列乘一行
full n by n matrix column times a row

542
00:34:24,429 --> 00:34:27,215
告诉我那个矩阵的特征向量
tell me an eigenvector of that matrix

543
00:34:27,220 --> 00:34:31,864
是的，如果我把你的矩阵乘以你
yes you if I multiply my matrix by you

544
00:34:31,869 --> 00:34:35,945
我得到了什么，我得到一些数字
I get what do I get I get some number

545
00:34:35,950 --> 00:34:40,435
时间U和那个数字是什么lambda
times U and what is that number lambda

546
00:34:40,440 --> 00:34:43,624
是lambda恰好是你的转置
is that lambda happens to be u transpose

547
00:34:43,629 --> 00:34:47,945
你和那是什么不同
U and what is the so that's different

548
00:34:47,950 --> 00:34:50,195
来自uutranspose这是一个矩阵
from uu transpose this was a matrix

549
00:34:50,200 --> 00:34:53,374
现在每个人都使用18超过65
everybody uses 18 over 6 5 now that's a

550
00:34:53,379 --> 00:34:56,404
号码，你能告诉我什么
number and what can you tell me about

551
00:34:56,409 --> 00:34:57,594
那个数字
that number

552
00:34:57,599 --> 00:35:03,515
它更大更好
it is greater well even more greater

553
00:35:03,520 --> 00:35:06,545
比0大，因为这是真的
than 0 greater because this is a true

554
00:35:06,550 --> 00:35:10,325
矢量所以这大于0
vector so this is greater than 0 it's

555
00:35:10,330 --> 00:35:12,035
所有其他的唯一特征值
the only eigenvalue all the other

556
00:35:12,040 --> 00:35:14,614
的那秩的特征值
eigenvalues of the of that of that rank

557
00:35:14,619 --> 00:35:17,405
一个矩阵是0但是一个非零
one matrix are 0 but the one non-zero

558
00:35:17,410 --> 00:35:18,365
本征
eigen

559
00:35:18,370 --> 00:35:21,005
价值在好的方面结束了它就是你
value is over on the plus side it's you

560
00:35:21,010 --> 00:35:23,945
转你，我们都意识到，作为
transpose you we all recognize that as

561
00:35:23,950 --> 00:35:26,975
U的长度肯定是平方的
the lengths of U squared it's certainly

562
00:35:26,980 --> 00:35:31,175
积极的，所以我们确实有一个积极的半
positive so we do have a positive semi

563
00:35:31,180 --> 00:35:34,225
确定矩阵你的猜测是什么
definite matrix what would your guess be

564
00:35:34,230 --> 00:35:39,875
对一个特征值的影响
of the effect on the eigenvalues of a so

565
00:35:39,880 --> 00:35:42,185
所以我回到我真正的问题我
so I'm coming back to my real problem I

566
00:35:42,190 --> 00:35:44,465
可以的值很抱歉是的我有一个
can values of s sorry yes I have a

567
00:35:44,470 --> 00:35:46,715
对称矩阵我说的是对称的
symmetric matrices I'm saying symmetric

568
00:35:46,720 --> 00:35:49,625
如果我有一个对称的话，你的猜测是什么？
what is your guess if I have a symmetric

569
00:35:49,630 --> 00:35:54,365
矩阵和我添加uutranspose做什么
matrix and I add on uu transpose what do

570
00:35:54,370 --> 00:35:56,935
你想象那对特征值有影响
you imagine that does to the eigenvalues

571
00:35:56,940 --> 00:36:00,295
你会说得对，就这么说吧
you're gonna get it right just say it

572
00:36:00,300 --> 00:36:03,305
s的特征值会发生什么
what happens to the eigenvalues of s if

573
00:36:03,310 --> 00:36:07,475
我加上uutranspose他们将会
I add on uu transpose they will they'll

574
00:36:07,480 --> 00:36:10,535
更积极的他们会上升这是一个
be more positive they'll go up this is a

575
00:36:10,540 --> 00:36:13,775
积极的事情就像添加17
positive thing it's like adding 17 to

576
00:36:13,780 --> 00:36:17,905
因此它会向上移动
something right it moves up so therefore

577
00:36:17,910 --> 00:36:26,285
我相信是这样的，现在我已经这样了
what I believe is so these now I have so

578
00:36:26,290 --> 00:36:29,255
是的我有两套特征值
yeah I've got two sets of eigenvalues

579
00:36:29,260 --> 00:36:33,335
现在一个是另一个的S参数
now one is the argument of s the other

580
00:36:33,340 --> 00:36:35,045
是特征值的不同
is the eigenvalues the different

581
00:36:35,050 --> 00:36:38,345
s的特征值所以我不能称之为
eigenvalues of s so what I can't call

582
00:36:38,350 --> 00:36:40,535
他们两个lambda表达式或我有麻烦所以
them both lambdas or I'm in trouble so

583
00:36:40,540 --> 00:36:45,395
让我让我开始，所以我会打电话给
let me let me start so I'll call the

584
00:36:45,400 --> 00:36:49,145
需要这样的另一个你最喜欢的
need so one other do you have a favorite

585
00:36:49,150 --> 00:36:51,545
其他希腊字母的特征值
other Greek letter for the eigenvalues

586
00:36:51,550 --> 00:36:57,905
只要你说一个，你的伽玛好吗？
of s gamma okay as long as you say a

587
00:36:57,910 --> 00:36:59,585
希腊字母我有一些想法如何
Greek letters I have some idea how to

588
00:36:59,590 --> 00:37:02,645
写Zeta在我看来就好像
write Zeta it seems to me like the

589
00:37:02,650 --> 00:37:05,345
世界上最难写的信和
world's toughest letter to write and

590
00:37:05,350 --> 00:37:07,985
电气工程师可以冷静地冲洗
electrical engineers can coolly flush

591
00:37:07,990 --> 00:37:13,295
阿尔法泽塔，但我从未成功过
alpha Zeta but like I've never succeeded

592
00:37:13,300 --> 00:37:16,325
所以你说的好吧
so so all right what did you say

593
00:37:16,330 --> 00:37:23,255
伽玛J原来没关系所以那些
gamma J of the original okay so those

594
00:37:23,260 --> 00:37:25,045
是原始的特征值
are the eigenvalues of the original

595
00:37:25,050 --> 00:37:28,315
这些是特征值
these are the eigenvalues of the

596
00:37:28,320 --> 00:37:31,505
修改过，我们期待着lambdas
modified and we're expecting the lambdas

597
00:37:31,510 --> 00:37:32,195
要大
to be big

598
00:37:32,200 --> 00:37:36,635
比起gammas哦，它只是一个
than the gammas oh it's just a

599
00:37:36,640 --> 00:37:39,005
定性陈述，这是真的
qualitative statement and it's true

600
00:37:39,010 --> 00:37:43,375
每个lambda都大于gamma哦
each lambda is bigger than the gamma oh

601
00:37:43,380 --> 00:37:47,675
对不起是啊是的每个lambda加入
sorry yeah yeah each lambda by adding

602
00:37:47,680 --> 00:37:49,804
这个东西，lambda比大
this stuff the lambdas are bigger than

603
00:37:49,809 --> 00:37:51,965
所以我就写的是lambda表达式
so I'll just write that lambdas are

604
00:37:51,970 --> 00:38:01,745
比gammas大，那是一个
bigger than gammas and that's a

605
00:38:01,750 --> 00:38:07,015
我们可以证明的基本事实
fundamental fact which we could prove

606
00:38:07,020 --> 00:38:10,354
而多了几分被称为当然的了
but a little more is known of course the

607
00:38:10,359 --> 00:38:14,614
问题是多大可以多大
question is how much bigger how much can

608
00:38:14,619 --> 00:38:18,604
他们做得更好，我不相信
they be way bigger well I don't believe

609
00:38:18,609 --> 00:38:21,215
他们可能会更大更多
they could be bigger by more than more

610
00:38:21,220 --> 00:38:24,965
比我自己的那个数字还有
than that number myself but there's

611
00:38:24,970 --> 00:38:28,235
那里有更好的消息
there's there's better news than that so

612
00:38:28,240 --> 00:38:30,094
lambdas比gammas大
the lambdas are bigger than the gammas

613
00:38:30,099 --> 00:38:34,625
所以lambda1大于gamma1所以
so lambda 1 is bigger than gamma 1 so

614
00:38:34,630 --> 00:38:38,854
这是SPlusuu
this is of this is the S Plus u u

615
00:38:38,859 --> 00:38:42,785
转置矩阵，这些是
transpose matrix and this these are the

616
00:38:42,790 --> 00:38:45,875
S矩阵λ1的特征值是
eigenvalues of the S matrix lambda 1 is

617
00:38:45,880 --> 00:38:48,125
大于伽玛1，但看起来是什么
bigger than gamma 1 but look what's

618
00:38:48,130 --> 00:38:52,475
发生在这一行文本中我是
happening in this line of text here I'm

619
00:38:52,480 --> 00:38:55,715
说λ2是λ2
saying that gamma 1 that lambda 2 is

620
00:38:55,720 --> 00:38:59,435
小于伽马1是不是工整
smaller than gamma 1 isn't that neat the

621
00:38:59,440 --> 00:39:06,155
特征值上升，但它们不仅仅是
eigenvalues go up but they don't just

622
00:39:06,160 --> 00:39:13,655
喜欢去任何地方，这就是所谓的
like go anywhere and that's called

623
00:39:13,660 --> 00:39:21,665
隔行扫描
interlacing

624
00:39:21,670 --> 00:39:24,045
所以这是其中一个很棒的
so this is one of those wonderful

625
00:39:24,050 --> 00:39:28,324
使你的心快乐的定理
theorems that makes your heart happy

626
00:39:28,329 --> 00:39:31,755
那个想法，如果我做一个等级
that that the idea that if I do a rank

627
00:39:31,760 --> 00:39:34,785
一个改变然后我和它是一个
one change then the I and it's a

628
00:39:34,790 --> 00:39:37,905
然后积极改变特征值
positive change then the eigen values

629
00:39:37,910 --> 00:39:42,135
增加，但他们不增加新的
increase but they don't increase the new

630
00:39:42,140 --> 00:39:44,715
特征值低于新的秒
eigen value is below the new second

631
00:39:44,720 --> 00:39:47,445
特征值它不会传递旧的
eigen value it doesn't pass up the old

632
00:39:47,450 --> 00:39:51,854
第一个特征值和新的第三个
first eigen value and the new third

633
00:39:51,859 --> 00:39:54,435
特征值不会过时
eigen value doesn't pass up the old

634
00:39:54,440 --> 00:39:57,735
第二个本征是的，是的，所以那是
second eigen does yeah yeah so that's

635
00:39:57,740 --> 00:40:01,025
这就是隔行扫描定理
the that's the interlacing theorem that

636
00:40:01,030 --> 00:40:04,875
与着名群众的名字相关联
associated with the names of famous mass

637
00:40:04,880 --> 00:40:08,054
那些人当然是你必须的
guys that's just of course you have to

638
00:40:08,059 --> 00:40:12,165
在我们写作的时候说这很漂亮
say that's beautiful while we're writing

639
00:40:12,170 --> 00:40:16,035
下来这样一个定理猜猜是什么
down such a theorem make a guess of what

640
00:40:16,040 --> 00:40:21,554
如果我做一个等级，定理就是
the theorem would be if I do a rank to

641
00:40:21,559 --> 00:40:28,834
改变假设我做了一个是对不起的
change suppose I do an A yes sorry s

642
00:40:28,839 --> 00:40:32,145
保持对称，我排名第一
staying symmetric and I do a rank one

643
00:40:32,150 --> 00:40:36,074
改变但是我也做了一个等级
change but then I also do a rank to

644
00:40:36,079 --> 00:40:41,715
改变说Ww转置好，所以是什么
change say W w transpose okay so what's

645
00:40:41,720 --> 00:40:44,864
这里的交易我对此有何了解
the deal here what do I know about that

646
00:40:44,869 --> 00:40:49,415
三角洲的变化矩阵在这里我
the change matrix the Delta s here I

647
00:40:49,420 --> 00:40:54,885
知道它的排名是2我假设你和
know it's rank is 2 I'm assuming you and

648
00:40:54,890 --> 00:40:56,834
W不是这样的
W were not in the same direction so

649
00:40:56,839 --> 00:41:00,824
这是一个2级矩阵，你可以做什么
that's a rank 2 matrix and what can you

650
00:41:00,829 --> 00:41:02,445
告诉我有关它的特征值
tell me about the eigenvalues of that

651
00:41:02,450 --> 00:41:07,574
秩2矩阵，因此它有n个特征值
Rank 2 matrix so it's got n eigenvalues

652
00:41:07,579 --> 00:41:10,005
因为它是N×n矩阵但是如何
because it's an N by n matrix but how

653
00:41:10,010 --> 00:41:14,975
它也有许多非零特征值
many nonzero eigenvalues has it got too

654
00:41:14,980 --> 00:41:18,975
因为它的等级是等级告诉的
because it's rank is to the rank tells

655
00:41:18,980 --> 00:41:20,715
你是非零特征值的数量
you the number of nonzero eigenvalues

656
00:41:20,720 --> 00:41:23,955
当矩阵是对称的时，它不会
when matrices are symmetric it doesn't

657
00:41:23,960 --> 00:41:26,114
告诉你，矩阵是否足够
tell you enough if matrices are

658
00:41:26,119 --> 00:41:28,955
非对称特征值可能很奇怪
unsymmetric eigenvalues can be weird so

659
00:41:28,960 --> 00:41:33,275
保持对称，这样就可以了
stay symmetric here ok so that that

660
00:41:33,280 --> 00:41:37,505
这有一个Chazz两个非零的特征值
this has a Chazz two nonzero eigenvalues

661
00:41:37,510 --> 00:41:39,955
你能告诉我他们的标志吗？
and can you tell me their sign

662
00:41:39,960 --> 00:41:42,695
什么是矩阵正面
what is that matrix positive

663
00:41:42,700 --> 00:41:45,425
当然，半定是的，它是
semi-definite yes of course it is of

664
00:41:45,430 --> 00:41:48,515
当然所以这是，而且这是和
course so this was and this was and

665
00:41:48,520 --> 00:41:50,405
肯定是
together it certainly is

666
00:41:50,410 --> 00:41:56,555
所以现在我增加了两个积极的排名
so now I've added a rank two positive

667
00:41:56,560 --> 00:41:58,715
确定正半正定矩阵
definite positive semi definite matrix

668
00:41:58,720 --> 00:42:00,965
现在我只是不打算重写
and now just I'm not going to rewrite

669
00:42:00,970 --> 00:42:03,395
此行，但你会期望
this line but what would you expect to

670
00:42:03,400 --> 00:42:08,705
是的，你会期待特征值
be true you would expect the eigenvalues

671
00:42:08,710 --> 00:42:13,885
增加但是lambda三有多大
increase but how big could lambda three

672
00:42:13,890 --> 00:42:19,205
很抱歉伽玛是的所以伽玛2让我们
are sorry gamma yeah so gamma 2 let's

673
00:42:19,210 --> 00:42:23,465
遵循伽玛2所以现在好吧也许我
follow gamma 2 so so now well maybe I

674
00:42:23,470 --> 00:42:26,135
应该使用希腊人的另外两个
should use another two the Greeks have

675
00:42:26,140 --> 00:42:30,085
除了lambda和gamma之外的任何其他字母
any other letters than lambda and gamma

676
00:42:30,090 --> 00:42:35,945
谁看到了这一切，谁知道，每个
who see all of that who knows that each

677
00:42:35,950 --> 00:42:39,935
一个我可以写alpha好的alpha是的
one I can write alpha good alpha yes

678
00:42:39,940 --> 00:42:43,535
alpha右边所以alpha作为特征alpha
alpha right so alpha as the eigen alpha

679
00:42:43,540 --> 00:42:50,225
是这个环的特征值
is the eigenvalues of this this ring to

680
00:42:50,230 --> 00:42:56,525
现在改变好我将成为什么样的人
change okay now what am I going to be

681
00:42:56,530 --> 00:43:00,455
能说我可以说些什么
able to say can I say anything about

682
00:43:00,460 --> 00:43:07,025
当然这很好，alpha1更大
this well of course alpha 1 is bigger

683
00:43:07,030 --> 00:43:09,755
比lambda1大于I
than lambda 1 which was bigger than I

684
00:43:09,760 --> 00:43:13,114
意思是我可以测量正确的上升我是
mean I can measure going up right I'm

685
00:43:13,119 --> 00:43:15,125
增加肯定或正面
adding positive definite or positive

686
00:43:15,130 --> 00:43:16,985
半定的东西，我无能为力
semi definite stuff there's no way I can

687
00:43:16,990 --> 00:43:18,935
伙计们可以开始贬低我了
guys can start going down on me

688
00:43:18,940 --> 00:43:23,135
所以alpha1大于或等于
so alpha 1 is greater or equal to the

689
00:43:23,140 --> 00:43:26,344
拉姆达1，它方才秩一
lambda 1 which had just a rank one

690
00:43:26,349 --> 00:43:28,775
变化大于等于
change which is greater equal to the to

691
00:43:28,780 --> 00:43:37,475
MU是mugammagamma1，它是
the MU was it mu gamma gamma 1 which is

692
00:43:37,480 --> 00:43:44,305
a，那是的，等等现在好了
a and that's yeah and so on okay now

693
00:43:44,310 --> 00:43:47,075
让我们看看alpha1是
let's see is alpha 1 is

694
00:43:47,080 --> 00:43:50,975
比我更大的阿尔法
one bigger than alpha what am i

695
00:43:50,980 --> 00:43:54,185
奋力在这里写下来好
struggling to to write down here well

696
00:43:54,190 --> 00:43:57,905
我能说什么才能说得好
what could I say well what can I say

697
00:43:57,910 --> 00:44:01,775
这反映了这一事实
that that reflects the fact that that

698
00:44:01,780 --> 00:44:08,495
这个lambda-没有抱歉
this lambda - no sorry

699
00:44:08,500 --> 00:44:12,155
它反映了伽玛一个上升的人
it reflects so gamma one went up who

700
00:44:12,160 --> 00:44:13,955
相机一个比lambda二大
camera one was bigger than lambda two

701
00:44:13,960 --> 00:44:16,535
伽玛一号就是这里的重点
that was the point here gamma one is

702
00:44:16,540 --> 00:44:18,605
更大，所以这很容易
bigger so so this was sort of easy

703
00:44:18,610 --> 00:44:21,245
因为我正在添加我期望的东西
because I'm adding stuff I expected the

704
00:44:21,250 --> 00:44:23,765
lambdas上去就是这个地方
lambdas to go up this is where the

705
00:44:23,770 --> 00:44:27,185
定理是它到目前为止没有上升
theorem is that it didn't go up so far

706
00:44:27,190 --> 00:44:30,935
至于传递或抱歉lambda二
as to pass or sorry the the lambda two

707
00:44:30,940 --> 00:44:34,345
上升没有超过伽玛1
which went up didn't pass up gamma 1

708
00:44:34,350 --> 00:44:37,975
lambda2现在没有通过gamma1
lambda 2 didn't pass up gamma 1 and now

709
00:44:37,980 --> 00:44:41,905
让我把这些话写下来吧
let me write those words down now the

710
00:44:41,910 --> 00:44:46,835
阿尔法2很好，它可以alpha
Alpha 2 well it could could alpha to

711
00:44:46,840 --> 00:44:53,785
传递lambda1和alpha3怎么样
pass up lambda 1 and what about alpha 3

712
00:44:53,790 --> 00:45:00,965
所以让我说出我相信的想法
so so let me say what I believe I think

713
00:45:00,970 --> 00:45:05,435
alpha2就像1落后但是我
alpha 2 which is like 1 behind but I'm

714
00:45:05,440 --> 00:45:09,425
添加排名-我认为alpha2可以通过
adding rank - I think alpha 2 could pass

715
00:45:09,430 --> 00:45:13,535
lambda1它可以通过lambda1但是
up lambda 1 it could pass lambda 1 but

716
00:45:13,540 --> 00:45:16,985
alpha3我不能相信alpha3
alpha 3 can't I believe that alpha 3 is

717
00:45:16,990 --> 00:45:21,485
比lambda1小，所以alpha3更小
smaller than lambda 1 so alpha 3 smaller

718
00:45:21,490 --> 00:45:26,065
然后伽玛不是原始的
than then then gamma wasn't the original

719
00:45:26,070 --> 00:45:37,445
是的好哦，无论如何我会把它弄好
yeah okay Oh anyway I'll get it right in

720
00:45:37,450 --> 00:45:40,565
你知道什么问题的笔记
the notes you you you know what question

721
00:45:40,570 --> 00:45:42,785
我问，这对我来说就是那个
I'm asking and that's for me that's the

722
00:45:42,790 --> 00:45:43,975
重要的事情
important thing

723
00:45:43,980 --> 00:45:49,325
现在有一个小问题，为什么
now there is a little matter of why is

724
00:45:49,330 --> 00:45:52,595
这是真的，这是好的情况让我
this true this is the good case let me

725
00:45:52,600 --> 00:45:54,665
给你另一个隔行扫描的例子
give you another example of interlacing

726
00:45:54,670 --> 00:45:58,265
我可以做到这就是真的
can I do that it's it's it it really

727
00:45:58,270 --> 00:46:00,725
来自这个，但让我给你一个
comes from this but let me give you an

728
00:46:00,730 --> 00:46:05,675
另一个例子就是这样我才引人注目
other example that's just striking so I

729
00:46:05,680 --> 00:46:10,925
有一个对称矩阵n由n调用它
have a symmetric matrix n by n call it s

730
00:46:10,930 --> 00:46:17,315
然后我扔掉了最后一排，并
and then I throw away the last row and

731
00:46:17,320 --> 00:46:21,275
列所以在这里是sn减1大
column so in here is s n minus 1 the big

732
00:46:21,280 --> 00:46:24,785
矩阵是SN，这是这个
matrix was SN the this is a this one is

733
00:46:24,790 --> 00:46:30,245
大小为n减1可以，所以它有排序
of size n minus 1 okay so it's got sort

734
00:46:30,250 --> 00:46:32,405
少自由度，因为
of less degrees of freedom because the

735
00:46:32,410 --> 00:46:35,845
自由的最后度得到去除，
last degree of freedom got removed and

736
00:46:35,850 --> 00:46:38,435
您如何看待特征值？
what do you think about the eigenvalues

737
00:46:38,440 --> 00:46:42,095
这是N减1的特征值
of this the N minus 1 eigenvalues of

738
00:46:42,100 --> 00:46:45,785
这和他们的n个特征值
this and the n eigenvalues of that they

739
00:46:45,790 --> 00:46:50,225
交织所以这个特征值也是如此
interlace so the eigenvalues so so this

740
00:46:50,230 --> 00:46:53,735
具有特征值lambda1
has eigen values lambda 1 this would

741
00:46:53,740 --> 00:46:57,055
具有小于该特征值的特征值
have an eigenvalue smaller than that

742
00:46:57,060 --> 00:47:00,935
这将有一个特征值lambda2
this would have an eigenvalue lambda 2

743
00:47:00,940 --> 00:47:05,105
这将使特征值更小
this would have an eigenvalue smaller

744
00:47:05,110 --> 00:47:07,565
比那样等等
than that and so on just the same

745
00:47:07,570 --> 00:47:09,635
隔行扫描，基本上是相同的
interlacing and basically for the same

746
00:47:09,640 --> 00:47:14,525
因为当你喜欢它的时候就像它一样
reason that when you like it's like it's

747
00:47:14,530 --> 00:47:18,575
这就是这个减小到n的大小
this is this this reduction to size n

748
00:47:18,580 --> 00:47:21,455
减1就像我说xn必须是
minus 1 is like I'm saying xn has to be

749
00:47:21,460 --> 00:47:23,915
能量为0或任何一个
0 in the energy or any any of those

750
00:47:23,920 --> 00:47:28,355
表达式和制作xn的事实
expressions and the fact of making xn be

751
00:47:28,360 --> 00:47:32,555
0是一个约束1
0 is a like a 1 constraint taking one

752
00:47:32,560 --> 00:47:35,315
它的自由度减少了
degree of freedom away it reduces the

753
00:47:35,320 --> 00:47:38,825
特征值但不是两个好现在我
eigen values but not by two okay now I

754
00:47:38,830 --> 00:47:45,905
有一个最后的神秘，让我尝试
have one final mystery and let me try to

755
00:47:45,910 --> 00:47:52,055
告诉你它让我担心的是什么
tell you what it worried me no what is

756
00:47:52,060 --> 00:47:55,565
它让我很担心
it that worried me yes

757
00:47:55,570 --> 00:48:02,395
假设你这就是假设这个
suppose this you that supposes this

758
00:48:02,400 --> 00:48:05,465
改变这种U该改变的是我
change this u this change that I'm

759
00:48:05,470 --> 00:48:10,145
假设它实际上是第二个
making suppose it's actually the second

760
00:48:10,150 --> 00:48:13,415
s的特征向量我也可以这样
eigenvector of s so so can I kind of

761
00:48:13,420 --> 00:48:17,365
写下来
write this down

762
00:48:17,370 --> 00:48:23,255
假设你实际上是第二个
suppose you is actually the second

763
00:48:23,260 --> 00:48:29,435
s的特征向量我的意思是这样的
eigenvector of s where I mean by that so

764
00:48:29,440 --> 00:48:33,005
我的意思是，有时U是lambda的2倍
I mean that s times U is lambda 2 times

765
00:48:33,010 --> 00:48:38,105
ü还好还好，现在我看，我要去
u okay okay now I look at and I'm gonna

766
00:48:38,110 --> 00:48:44,915
改变它s+uu转置那是什么
change it s + uu transpose that's what

767
00:48:44,920 --> 00:48:46,625
我一直在看，这会让人感动
I've been looking at and that moves the

768
00:48:46,630 --> 00:48:49,535
特征值上升但令我担心的是
eigenvalues up but what worries me is

769
00:48:49,540 --> 00:48:55,355
就好像我把它乘以20大一些
like if I multiply this by 20 some big

770
00:48:55,360 --> 00:48:59,125
我想要移动那个特征值
number I'm gonna move that eigenvalue

771
00:48:59,130 --> 00:49:03,995
过去的方式我就像我担心的那样
way up way past I'm like I got worried

772
00:49:04,000 --> 00:49:07,265
关于这个不平等的权利
about this this inequality right

773
00:49:07,270 --> 00:49:11,855
如果我，但当我添加这就是那个
if I but when I add this that's that

774
00:49:11,860 --> 00:49:20,345
相同的U是lambda2加20你写su
same U is lambda 2 plus 20 you write su

775
00:49:20,350 --> 00:49:26,735
是lambda2u，这20是20U是
is lambda 2 u and this 20 is 20 U is a

776
00:49:26,740 --> 00:49:30,515
单位矢量所以是的你在这里看到我的担心
unit vector so yes you see my worry here

777
00:49:30,520 --> 00:49:35,465
我正在进行等级1更改，但它已被移动
I'm doing a Rank 1 change but it's moved

778
00:49:35,470 --> 00:49:39,755
在特征值方式上如此可能
in eigen value way way up so how could

779
00:49:39,760 --> 00:49:44,945
这种说法是真的，这种危险
this danger this this statement be true

780
00:49:44,950 --> 00:49:47,735
所以我只是我只是想出了这里
so I'm just I've just figured out here

781
00:49:47,740 --> 00:49:51,425
什么配得好呀
what what gamete well yeah

782
00:49:51,430 --> 00:49:54,995
你看到我的问题，我可以把它留下来
you see my question I could leave it as

783
00:49:55,000 --> 00:49:58,085
下次回答的问题让我这样做
a question to answer next time let me do

784
00:49:58,090 --> 00:50:03,215
那个，我会把它放在网上，所以你会
that and I'll put it on online so you'll

785
00:50:03,220 --> 00:50:06,935
看清楚它看起来像它
see it clearly it looks like and it it

786
00:50:06,940 --> 00:50:11,135
发生这个特征向量现在有
happens that this eigen vector now has

787
00:50:11,140 --> 00:50:14,315
特征值lambda2加20为什么不
eigen value lambda 2 plus 20 why doesn't

788
00:50:14,320 --> 00:50:19,385
吹走这句话好吧，我会的
that blow away this statement okay I'll

789
00:50:19,390 --> 00:50:22,085
把它放在一边，因为它有点来了
put that because it's sort of coming

790
00:50:22,090 --> 00:50:24,905
与零下10秒，在去
with with minus 10 seconds to go in the

791
00:50:24,910 --> 00:50:28,245
类，以便让我们离开那
class so so let's leave that

792
00:50:28,250 --> 00:50:30,555
下次再讨论这个问题
a discussion of this for next time and

793
00:50:30,560 --> 00:50:34,035
那么如果我对这个讲座感到满意的话
then but I'm happy with this lecture if

794
00:50:34,040 --> 00:50:37,695
你是我使用V的最后一次演讲
you are the last lecture I got using V's

795
00:50:37,700 --> 00:50:41,085
混淆了，我在这里不可靠
mixed up and it's not reliable here I

796
00:50:41,090 --> 00:50:46,089
like the proof of D lambda DT and we're

797
00:50:41,090 --> 00:50:50,560
就像DlambdaDT的证明一样，我们也是

798
00:50:46,099 --> 00:50:50,560
从这个话题开始-非常感谢你
started on this topic - good thank you