18-Counting Parameters in SVD, LU, QR, Saddle Points.srt

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visit MIT opencourseware at ocw.mit.edu

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好的，所以我想我今天就开始了
ok so I thought I'd begin today with as

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我们要来排序的结束
we're coming to the end of the sort of

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专注于线性代数和移动到
focus on linear algebra and moving on to

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一点点概率
a little probability a a little more

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优化和很多深刻的
optimization and a a lot of deep

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00:00:40,430 --> 00:00:43,815
系统的学习，这是像我的方式
learning so this was like my way of

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回顾写下大事
review to write down the big

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矩阵的因式分解等我的
factorizations of a matrix and so my

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想法，它有点喜欢它
idea and it's kind of enjoyed it as

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检查一些免费的
checking that a number of free

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参数在L＆U或q和r或中表示
parameters say in L&U or in q and r or

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每个人的每一个人数
every each of those that the number of

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自由参数与数字一致
free parameters agrees with the number

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其中的参数如N平方
of parameters in a itself like N squared

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通常所以通常有N平方和
usually so a usually has N squared and

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然后我们可以在我们之后更换一个if
then can we replace a if after we've

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计算你可以扔掉一个是的
computed Ln you can we throw away a yes

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因为所有信息都在L和
because all the information is in L and

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U和它用n矩阵填充相同的n
U and it fills that same n by n matrix

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这很明显，因为L是
well that's kind of obvious because L is

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下三角形和对角线全部
a lower triangular and the diagonal all

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那些没有不自由参数
ones is not are not free parameters and

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U是三角形的上三角形和
U is triangular upper triangular and

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这是自由的枢轴
it's diagonal the pivots those are free

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这样的参数，但我可以写
parameters so that but can I just write

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倒数，所以我会经历
down the count so so I'll go through

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我这之后很快就完成了这些
each of these just quickly after I've

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00:02:00,619 --> 00:02:03,974
弄清楚这些是怎样的
figured out how these are the sort of

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00:02:03,979 --> 00:02:07,754
这个积木有多少免费
the building blocks so how many free

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这两个参数都在那里
parameters are there in these two

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我觉得三角矩阵很好
triangular matrices well I think the

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回答
answer

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1/2和n减1和1/2以及n加1
1/2 and n minus 1 and 1/2 and n plus 1

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这是一个熟悉的数字，是一些
that's a familiar number that's the some

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你认识到这是1的总和
of you recognize that as the sum of 1

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加2到n，你有一个免费/
plus 2 up to n and you have one free /

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在上三角你
in the in the upper triangular you

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你有一个免费参数
you've got one free parameter up in the

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在下一个角落2和你一样
corner 2 in the next one and as you're

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下来你最终得到了n
coming down you end up with n on the

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主对角线，他们加起来
main diagonal and they add up to that

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00:02:47,940 --> 00:02:51,234
而且你看到那两个是不同的
and you see that those two are different

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由n这是我们想要的好
by n which is what we want okay

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对比答案明显如何
diagonal the answers obviously in how

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关于特征向量矩阵，所以这是
about the eigenvector matrix so this is

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00:03:05,610 --> 00:03:08,575
这整个练习就像是一件事
this whole exercise is like something

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我从来没有在教科书中看过
I've never seen in a textbook in and

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它只是但对我来说带回所有
it's just but for me it brings back all

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这些关键真的是浓缩的过程
these key really the condensed course in

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线性代数是在该顶线，使得
linear algebra is on that top line so

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一个多少个免费参数
how many free parameters in an

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特征向量矩阵当然可以
eigenvector matrix okay and of course

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你是在想什么是规则
you're sort of thinking what's the rule

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对于免费参数所以我的答案是
for free parameters so the my answer is

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将是免费的数量
going to be for the number of free

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参数，所以这是一个N乘n矩阵
parameters so this is an N by n matrix

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其中有n个特征向量但是
with the n eigenvectors in it but

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那里有一定的自由和什么
there's a certain freedom there and what

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是我们拥有的自由
is that what freedom do we have in

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每个选择特征向量矩阵
choosing the eigenvector matrix every

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特征向量可以乘以a
eigen vector can be multiplied by a

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标量如果x是特征向量，则为2x
scalar if x is an eigenvector so is 2x

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00:04:04,109 --> 00:04:07,555
所以是3倍，所以我们可以做到
so is 3x so we could we could make a

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会议第一部分是
convention that the first component was

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总是1也许不会是最多的
always 1 maybe that wouldn't be the most

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世界上的智能会议但是
intelligent convention in the world but

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它会显示那一行的顶行
it would show that that top row of ones

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00:04:18,739 --> 00:04:22,164
不算数，所以我得到N.
were not to be counted so I get N

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00:04:22,169 --> 00:04:26,724
平方减去那个哦，好吧
squared minus n for that oh yeah well

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做了那些
having done those

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-让我让我看看这个
- let me let me look at this at this one

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不说出来，总那么平方
does that come out a total then squared

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是的，因为特征向量x有N.
yes because the eigenvector x has N

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平方减去N乘这个推理小
squared minus n by this reasoning little

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hokey推理我刚才给的和
hokey reasoning that I just gave and

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然后还有更多的特征值
then there n more for the eigenvalue

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矩阵，没有什么可留给的
matrix and there's nothing left for the

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因为它是本征的逆
eigen the inverse because it's

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由x确定所以你看到计数
determined by x so do you see the count

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现在我加起来为n平方
adding up to n squared for those now I

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开着正交一个我觉得我们
left open the orthogonal one I think we

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00:05:03,390 --> 00:05:05,785
那种在谈论期间谈到的那种
kind of talked about that during the

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当我们遇到它时，它会少一些
when we met it and it's a little less

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显而易见，但你还记得我是吗？
obvious but do you remember so I'm

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谈论N乘n正交
talking about an N by n orthogonal

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矩阵Q所以有多少自由参数
matrix Q so how many free parameters in

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第一列，那列是我们的
column one that that column is what we

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总是打电话给Q1它是否有免费结束
always call Q 1 does it have end free

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参数或是否有条件
parameters or is there a condition that

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削减那些有一个条件
that cuts that back there is a condition

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对，什么是条件
right and what's the condition on the

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00:05:35,310 --> 00:05:38,365
删除一个的第一列
first column that that removes one

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参数它的标准化它的长度是
parameter it's normalized it's length is

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1所以我只从第一个得到n减1
1 so I only get n minus 1 from the first

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专栏，现在如果我转移到
column and now if I move over to the

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第二列有多少免费参数
second column how many free parameters

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00:05:53,490 --> 00:05:57,655
再一次，这是一个单位矢量，但也
there again it's a unit vector but also

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它与第一个正交到两个正交
it is orthogonal to the first so to two

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参数得到了两个强加的规则
parameters got you two rules got imposed

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00:06:06,360 --> 00:06:09,235
和两个参数得到去除，这
and two parameters got removed so this

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是n减2然后最后是什么
is n minus 2 and then finally whatever

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所以我认为那些家伙
so I think that that some of these guys

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00:06:15,479 --> 00:06:17,905
和我们在这里完全一样
is exactly the same that we had up here

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00:06:17,910 --> 00:06:23,005
我认为它也是1/2和n减1或
I think it's also 1/2 and n minus 1 or

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00:06:23,010 --> 00:06:27,775
1/2平方-亚当是的，所以不是
1/2 n squared - Adam yeah yeah so not as

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00:06:27,780 --> 00:06:30,565
你可能认为很多就像矩阵一样
many as you might think as the matrix is

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00:06:30,570 --> 00:06:34,395
大小N平方现在可以使用那些
size N squared now can I use those

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因为这些就像建筑物一样
because these are the like the building

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块我可以看看这些让我们看看
blocks can I just check these let's see

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沿着清单L次你这么L
just go along the list L times u so L

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有这个，你有，当我添加
had this and you had that and when I add

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00:06:47,820 --> 00:06:51,235
那些它加起来N平方正确的
those it adds up to N squared right the

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减去取消加号和半个N.
minus cancels the plus and the half N

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平方两次给我N平方那么好
squared twice gives me N squared so good

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那个
for that one

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00:06:58,290 --> 00:07:03,025
那么QR井R是上层
what about Q R well R is upper

125
00:07:03,030 --> 00:07:08,395
三角形如此然后Q哦是的我们
triangular like so and then Q oh yeah we

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00:07:08,400 --> 00:07:11,485
只是在那里得到它所以Q次R
just got it right there so for Q times R

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00:07:11,490 --> 00:07:15,685
这就是Plus再次加入N.
it's that Plus that again adding to N

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00:07:15,690 --> 00:07:19,315
对那个平方很好并且平方
squared good for that one and squared

129
00:07:19,320 --> 00:07:23,845
对于一个这一次，我们只是做了ñ
for that one this one we just did N

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00:07:23,850 --> 00:07:27,055
平方减去N和X并且在
squared minus N and X and on the

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00:07:27,060 --> 00:07:30,055
对角线总N平方怎么样
diagonal total N squared what about this

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00:07:30,060 --> 00:07:35,575
关于人的大O真正
guy what about the Big O really

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00:07:35,580 --> 00:07:37,975
我通常会这样做的基本原则
fundamental one that I would normally

134
00:07:37,980 --> 00:07:40,825
将矩阵写为s而不是a
write the matrix as s instead of a to

135
00:07:40,830 --> 00:07:44,515
提醒我们，这里的矩阵
remind us that it that the matrix here

136
00:07:44,520 --> 00:07:50,305
是对称的，所以我不期待看到
is symmetric so I'm not expecting ensk

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00:07:50,310 --> 00:07:53,545
我应该拥有对称的地方
where'd for a symmetric me I should have

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00:07:53,550 --> 00:07:54,685
把它放在我的名单上
put that on my list

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00:07:54,690 --> 00:08:00,195
对称矩阵的计数是多少
what's the count for a symmetric matrix

140
00:08:00,200 --> 00:08:04,975
在这里，所以我不期待得到N.
s here so I'm not expecting to get N

141
00:08:04,980 --> 00:08:07,315
平方我只是期待得到
squared I'm only expecting to get the

142
00:08:07,320 --> 00:08:13,285
对称的数量是多少
number of symmetric s what's the number

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00:08:13,290 --> 00:08:15,505
自由参数，我会是我
of free parameters that I would that I

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00:08:15,510 --> 00:08:19,195
从那开始我希望会重新出现
start with that I hope will reappear in

145
00:08:19,200 --> 00:08:25,345
Q和lambda有多少是什么交易
Q and lambda so how many what's the deal

146
00:08:25,350 --> 00:08:30,325
对于一个对称矩阵让我们看看我
for a symmetric matrix let's see I'm

147
00:08:30,330 --> 00:08:33,775
自由选择是相同的计数为
free to choose is it the same count as

148
00:08:33,780 --> 00:08:37,164
这是的，因为我可以自由地选择
this yeah because I'm free to choose the

149
00:08:37,169 --> 00:08:39,925
上三角部分和对角线
upper triangular part and the diagonal

150
00:08:39,930 --> 00:08:42,685
但我不能自由选择较低的
but I'm not free to choose the lower so

151
00:08:42,690 --> 00:08:46,585
所以我说它是n=1的1/2n倍
so I'd say it's 1/2 n times n minus 1

152
00:08:46,590 --> 00:08:53,545
并且加上在那里确定，所以对不起对角线
and plus sorry diagonals in there ok so

153
00:08:53,550 --> 00:08:55,255
我明白了吗
do I get that

154
00:08:55,260 --> 00:09:01,365
这些家伙的N平方加一半加n
half of N squared plus n from these guys

155
00:09:01,370 --> 00:09:05,995
好吧，我可能做对角线的人
well I probably do the diagonal guy

156
00:09:06,000 --> 00:09:09,505
让我在这给了我，那是一个
gives me in this gives me n and that's a

157
00:09:09,510 --> 00:09:13,165
Q这是我最喜欢的号码
Q which is my other favorite number

158
00:09:13,170 --> 00:09:18,805
在那里，当我把它添加到那个
there and when I add that to that that

159
00:09:18,810 --> 00:09:23,155
变成一个加号，我好耶
becomes a plus sign and I'm good yeah

160
00:09:23,160 --> 00:09:26,635
你知道我喜欢这样做但是
you see how I enjoy doing this right but

161
00:09:26,640 --> 00:09:28,665
我快要结束了，但是最后一个
I'm near the end but the last one it's

162
00:09:28,670 --> 00:09:34,785
有点不太知名好的Q时间说
kind of not well known okay Q time says

163
00:09:34,790 --> 00:09:37,255
你还记得那个因素分解
do you remember that factorization

164
00:09:37,260 --> 00:09:39,435
这就是所谓的极性分解
that's a called the polar decomposition

165
00:09:39,440 --> 00:09:43,105
这是一个正交时间a
it's a it's an orthogonal times a

166
00:09:43,110 --> 00:09:46,885
对称的，经常用于
symmetric and it is often used in

167
00:09:46,890 --> 00:09:52,155
工程作为一种分解方法
engineering as a way to decompose a

168
00:09:52,160 --> 00:09:57,955
无论如何，位移或应变矩阵Q.
displacement or strain matrix anyway Q

169
00:09:57,960 --> 00:10:00,745
倍于此，它实际上是非常的
times this and it actually it is very

170
00:10:00,750 --> 00:10:04,105
非常接近SVD，我有朋友
very close to the SVD and I have friends

171
00:10:04,110 --> 00:10:06,925
谁更好地计算Qs而不是
who say better to compute Q s than the

172
00:10:06,930 --> 00:10:11,005
SVD然后只是继续前进Q.
SVD and then just move along anyway Q

173
00:10:11,010 --> 00:10:18,505
这个时候Q就是这个人，而且是什么
times this so Q is this guy and s what's

174
00:10:18,510 --> 00:10:23,325
对称的是这个家伙
s symmetric that's this guy

175
00:10:23,330 --> 00:10:27,145
所以，Q让我写那封信Q.
so that's Q let me write that letter Q

176
00:10:27,150 --> 00:10:29,755
所以我不会失去它
and s so I don't lose it

177
00:10:29,760 --> 00:10:35,265
那些是什么加起来和平方
what are those add up to and squared

178
00:10:35,270 --> 00:10:41,305
快乐好，所以最后SVD友好了
happy ok so finally the SVD friendly the

179
00:10:41,310 --> 00:10:44,035
SVD是什么计数
SVD what's the count

180
00:10:44,040 --> 00:10:45,925
好吧现在我不知道我有
well now I don't know I've got

181
00:10:45,930 --> 00:10:48,895
那里有长方形的东西，所以我有
rectangular stuff in there so I've got

182
00:10:48,900 --> 00:10:52,975
我准备好了这个，我必须这样做
to I'm ready for this one and I have to

183
00:10:52,980 --> 00:10:55,675
觉得有点
think a little bit

184
00:10:55,680 --> 00:11:01,345
我们可能已经这样做了所以我拥有了
and we may have done this so I have I

185
00:11:01,350 --> 00:11:06,265
让我们假设M等于M
let's suppose that M is lesser equal M

186
00:11:06,270 --> 00:11:11,525
假设它是，否则我们
suppose that it is yeah otherwise we

187
00:11:11,530 --> 00:11:14,045
只会转置并查看SVD
would just transpose and look at the SVD

188
00:11:14,050 --> 00:11:17,675
所以让我们说M小于或等于n所以让我们来吧
so let's say M less or equal n so let's

189
00:11:17,680 --> 00:11:22,745
说它有完整的排名，是什么
say it's got full rank and what's the

190
00:11:22,750 --> 00:11:26,735
矩阵可以具有M的最大等级
largest rank that the matrix can have M

191
00:11:26,740 --> 00:11:32,135
显然满级M所以SVD将是M.
clearly full rank M so the SVD will be M

192
00:11:32,140 --> 00:11:37,745
由M让我们记住你的
by M let's remember the the you the

193
00:11:37,750 --> 00:11:41,675
Sigma和V转置这将是M
Sigma and the V transpose this will be M

194
00:11:41,680 --> 00:11:46,295
通过n，这将是n乘以n
by n and this will be n by n for the

195
00:11:46,300 --> 00:11:49,205
满刻度这个VD，如果排名是
full scale this VD and if the rank is

196
00:11:49,210 --> 00:11:55,235
等于M然后我真的希望得到我
equal to M then I really expect to get I

197
00:11:55,240 --> 00:11:58,885
期望它加起来总和
expect it to add up to the total for a

198
00:11:58,890 --> 00:12:06,875
对于原始的a有MN的权利
for a the original a has MN right it's

199
00:12:06,880 --> 00:12:14,395
在M×n矩阵中，矩阵a是M乘以n
an M by n matrix the matrix a is M by n

200
00:12:14,400 --> 00:12:18,305
与他们在给予我的方面较少或相等
with them less or equal in giving me

201
00:12:18,310 --> 00:12:25,865
这些东西也有MN参数这样做
these things so has MN parameters so do

202
00:12:25,870 --> 00:12:30,005
我们得到M倍n在此，我希望我们做的
we get M times n from this I hope we do

203
00:12:30,010 --> 00:12:33,455
我知道我们从Sigma获得了多少
I know how many we get from Sigma what

204
00:12:33,460 --> 00:12:37,075
什么是SigmaM的数量
how many what's the count for a Sigma M

205
00:12:37,080 --> 00:12:42,305
什么是V的计数，所以这是一个
and what's the count for V so that's an

206
00:12:42,310 --> 00:12:46,775
Nbyn，你的数量还算什么呢
N by n and what's the count for you okay

207
00:12:46,780 --> 00:12:49,715
是的，他们是正交矩阵，所以
yeah they're orthogonal matrices so that

208
00:12:49,720 --> 00:12:52,025
我应该能够达到那条路线
I should be able to go up to that line

209
00:12:52,030 --> 00:12:55,535
这是MbyMone，所以就是这样
this was an M by M one so that's is that

210
00:12:55,540 --> 00:12:59,135
一半和一个减去我复制那个
a half and n minus one am i copying that

211
00:12:59,140 --> 00:13:02,275
正确了这个圈子的存在
correctly out of this circle there

212
00:13:02,280 --> 00:13:05,705
那就是M乘M正交
that's that's an M by M orthogonal

213
00:13:05,710 --> 00:13:07,415
矩阵哦
matrix Oh

214
00:13:07,420 --> 00:13:09,875
但我必须写出来，这是愚蠢的
but I have to write in that was foolish

215
00:13:09,880 --> 00:13:15,695
好吧，他们是啊因为那个矩阵
okay em em yeah because that matrix

216
00:13:15,700 --> 00:13:21,035
这是一个大小M，所以这是一个M然后我
it's a size M so that's an M and then I

217
00:13:21,040 --> 00:13:25,745
有那个，然后我有任何V
have that and then I have whatever V

218
00:13:25,750 --> 00:13:29,915
转移结束n哦什么是交易
transpose end by n oh what's the deal in

219
00:13:29,920 --> 00:13:32,875
嗯
there hmm

220
00:13:32,880 --> 00:13:41,905
我想要所有1/2+n减1哦
do I want all of the 1/2 + n minus 1 Oh

221
00:13:41,910 --> 00:13:46,105
上帝，我以为我会直截了当
God I thought I'd got this straight

222
00:13:46,110 --> 00:13:52,175
他知道我的意思是我可以减去
does he know I meant I could subtract

223
00:13:52,180 --> 00:13:54,905
从这个，并找出我
this from this and and find out what I

224
00:13:54,910 --> 00:13:57,295
应该说哇
should say whoa

225
00:13:57,300 --> 00:14:00,685
众所周知，学生也会这样做
students have been known to do this too

226
00:14:00,690 --> 00:14:04,625
让我们看看好吧，无论如何都要试着去思考
let's see well let's try to think anyway

227
00:14:04,630 --> 00:14:08,645
所以我有这个n由n提交n
so I have this n by n submit this n by n

228
00:14:08,650 --> 00:14:14,105
正交矩阵，所以它是第一个它
orthogonal matrix so it's first it could

229
00:14:14,110 --> 00:14:19,375
它可以是任何正交矩阵是的
it could be any orthogonal matrix yeah

230
00:14:19,380 --> 00:14:24,575
但它只是前M列
but is it only the first M columns that

231
00:14:24,580 --> 00:14:27,845
我真的需要休息，我可以
I really need the rest I could just

232
00:14:27,850 --> 00:14:33,065
扔掉所以我让我试着想象
throw away so I let me try to imagine

233
00:14:33,070 --> 00:14:35,915
它只是我的第一口井
that it's just the first well then I

234
00:14:35,920 --> 00:14:39,905
在这里不会有任何结局，所以也许我
won't have any end in here so maybe I

235
00:14:39,910 --> 00:14:45,275
最好拿1/2，你知道帮助哦哦
better take 1/2 and you know help oh oh

236
00:14:45,280 --> 00:14:46,895
当然是
yes of course

237
00:14:46,900 --> 00:14:52,435
哈，我只有M列才重要
ha I've got only M columns that matter

238
00:14:52,440 --> 00:14:56,645
现在每个人都喜欢的诱饵
the bait the everybody sort of now

239
00:14:56,650 --> 00:14:59,795
了解SVD我们是对的
understands that SVD we've got the right

240
00:14:59,800 --> 00:15:03,485
等级是我不要忘了那么好
the rank is am don't forget that ok then

241
00:15:03,490 --> 00:15:07,705
第一个是第一个M
the first the the first are the first M

242
00:15:07,710 --> 00:15:10,895
的V列是很重要的是
columns of V are important those are the

243
00:15:10,900 --> 00:15:13,955
非零的奇异向量
singular vectors that go with nonzero

244
00:15:13,960 --> 00:15:17,405
真正重要的奇异值
singular values that really matter so

245
00:15:17,410 --> 00:15:19,895
那是什么，其余的并不重要
that's and the rest really don't matter

246
00:15:19,900 --> 00:15:21,185
所以
so

247
00:15:21,190 --> 00:15:24,425
所以我只想知道如何
so I'm going to just I have to count how

248
00:15:24,430 --> 00:15:29,345
很多，所以抱歉V重要
many so here sorry V the the important

249
00:15:29,350 --> 00:15:38,555
你的一部分有多少只有M
part of thee has has how many and only M

250
00:15:38,560 --> 00:15:42,125
列，但它是一个N乘n矩阵和
columns but it's an N by n matrix and

251
00:15:42,130 --> 00:15:44,585
那些列是正交的，所以很荣幸
those columns are orthogonal so honor

252
00:15:44,590 --> 00:15:47,675
答案是不MN这家伙的我
the answer is not MN for this guy its I

253
00:15:47,680 --> 00:15:49,895
必须经历这种愚蠢
have to go through this foolish

254
00:15:49,900 --> 00:15:55,715
再次推理我有n减1加n
reasoning again I have n minus 1 plus n

255
00:15:55,720 --> 00:16:05,315
减2加上等加n减M
minus 2 plus so on plus n minus M there

256
00:16:05,320 --> 00:16:07,415
在第一个中是n减1个参数
were n minus 1 parameters in the first

257
00:16:07,420 --> 00:16:09,785
在思想哦矢量或单位矢量
over thought oh vector or a unit vector

258
00:16:09,790 --> 00:16:12,215
n减去2，在第二个到n
n minus 2 in the second one up to n

259
00:16:12,220 --> 00:16:15,725
在第三个减去M然后V有一些
minus M in the third and then V has some

260
00:16:15,730 --> 00:16:18,305
更多真正来自的专栏
more columns that are coming really from

261
00:16:18,310 --> 00:16:23,075
这是不是一个空的空间都没有
a null space that that are not are not

262
00:16:23,080 --> 00:16:24,095
重要
important

263
00:16:24,100 --> 00:16:28,825
我相信这是正确的做法
I believe this is the right thing to do

264
00:16:28,830 --> 00:16:33,755
我希望你同意，现在我的意思
I'm hoping you agree and now I I mean

265
00:16:33,760 --> 00:16:35,735
我希望那些更多的东西
I'm hoping even more that those that up

266
00:16:35,740 --> 00:16:40,265
好几次，我有一半和
to M times in okay I have a half and

267
00:16:40,270 --> 00:16:43,165
广场他们必须完成这件事
square they have to total this thing

268
00:16:43,170 --> 00:16:47,165
好吧，这有mm条款
okay this had m m terms

269
00:16:47,170 --> 00:16:52,925
所以有这些目的，然后我
so there's m m of these ends and then I

270
00:16:52,930 --> 00:16:55,085
必须减去一加二加
have to subtract off one plus two plus

271
00:16:55,090 --> 00:16:58,385
三个到M，所以我是什么
three up to M and so what am i

272
00:16:58,390 --> 00:17:01,685
减去一些人的东西
subtracting off what's that some one

273
00:17:01,690 --> 00:17:07,204
加上两个加三个停在M的位置
plus two plus three stopping at M it's

274
00:17:07,209 --> 00:17:11,194
其中一个半人是1/2毫米
one of these guys one-half is it 1/2 mm

275
00:17:11,199 --> 00:17:17,975
再加上一个1/2米米加1个对不起1/2米
plus one yeah 1/2 m m plus 1 sorry 1/2 m

276
00:17:17,980 --> 00:17:22,265
m加1我应该喜欢这个
m plus 1 I would supposed to enjoy this

277
00:17:22,270 --> 00:17:27,425
而且现在有点紧张但是没关系，所以我
and now little nervous but okay so I

278
00:17:27,430 --> 00:17:29,425
相信那
believe that that

279
00:17:29,430 --> 00:17:32,605
是的，我们有明确的MN
is that okay well we have the MN that's

280
00:17:32,610 --> 00:17:36,205
这是一个很好的迹象，我们正在拍摄
a good sign that we're shooting for so

281
00:17:36,210 --> 00:17:39,175
它的其余部分是否会增加
does the rest of it add to nothing well

282
00:17:39,180 --> 00:17:42,835
我猜是的，我猜我放的时候会这样
I guess yeah I guess it does when I put

283
00:17:42,840 --> 00:17:46,465
这两个在一起我有1/2毫米加
these two together I have 1/2 mm plus

284
00:17:46,470 --> 00:17:48,985
一个然后我将它减去
one and then I'm subtracting it away

285
00:17:48,990 --> 00:17:50,815
再来，所以我得到了阿门
again so I get amen

286
00:17:50,820 --> 00:17:58,435
好吧，我们什么都不会
all right well we wouldn't anything

287
00:17:58,440 --> 00:18:00,895
在我举起董事会并委托之前
before I raise that board and consign

288
00:18:00,900 --> 00:18:03,775
对历史来说，我应该停下来
that to history is there should I pause

289
00:18:03,780 --> 00:18:07,405
多一点分钟，这将是像
a little more minute this will be like

290
00:18:07,410 --> 00:18:09,975
我希望有一页的附录
I'm hoping a one-page appendix to the

291
00:18:09,980 --> 00:18:13,105
笔记和书，你会看到它但是
notes and the book and you'll see it but

292
00:18:13,110 --> 00:18:18,265
我有一个多计数做，然后
I do have one more count to do and then

293
00:18:18,270 --> 00:18:22,315
我对这篇评论很满意并准备好了
I'm good with this review and ready to

294
00:18:22,320 --> 00:18:26,005
继续前进到马鞍的话题
move onward to the topic of saddle

295
00:18:26,010 --> 00:18:28,795
积分并准备继续前进
points and ready to move onward after

296
00:18:28,800 --> 00:18:30,655
那好，我会说大约一点点
that well I'll say a little bit about

297
00:18:30,660 --> 00:18:35,065
我正在创建的下一个实验作业
the next lab homework that I'm creating

298
00:18:35,070 --> 00:18:39,535
然后我们的下一个主题将是
and then our next topic will be like

299
00:18:39,540 --> 00:18:42,595
协方差矩阵有点
covariance matrices it a little

300
00:18:42,600 --> 00:18:46,285
本周统计我们和我们
statistics this week and then we and we

301
00:18:46,290 --> 00:18:49,195
得到一个星期的假，我们能来消化它，
get a week off we could to digest it and

302
00:18:49,200 --> 00:18:54,445
然后回来进行梯度下降和
then come back for gradient descent and

303
00:18:54,450 --> 00:18:58,515
深度学习和那些事情还可以
deep learning and and those things okay

304
00:18:58,520 --> 00:19:01,555
每个人都对此很满意，那么我的是什么
everybody happy with that so what's my

305
00:19:01,560 --> 00:19:05,665
最后一个问题我的最后一个问题是
final question my final question is the

306
00:19:05,670 --> 00:19:20,905
SVD用于任何秩R的任何矩阵，从而
SVD for for any any matrix of rank R so

307
00:19:20,910 --> 00:19:26,545
它是一个M乘n矩阵，但等级是
it's an M by n matrix but the rank is

308
00:19:26,550 --> 00:19:28,544
只有R.
only R

309
00:19:28,549 --> 00:19:31,274
这是一个自然的问题多少
it's a natural question how many

310
00:19:31,279 --> 00:19:35,445
参数存在于排名为R的a中
parameters are there in a in a ranked R

311
00:19:35,450 --> 00:19:39,764
矩阵，这可能是我们甚至可能有
matrix and that may be we may even have

312
00:19:39,769 --> 00:19:42,794
谈到这个问题，我有两个
touched on this question and I have two

313
00:19:42,799 --> 00:19:46,654
回答它的方法和一种方法是SVD
ways to answer it and one way is the SVD

314
00:19:46,659 --> 00:19:50,024
所以，让我和那个类似于什么
so let me and that'll be similar to what

315
00:19:50,029 --> 00:19:53,414
如果排名是，我只是推到那里
I just pushed up there so if the rank is

316
00:19:53,419 --> 00:19:57,825
我们这个典型排名的SVD我们的
our the SVD of this typical rank our

317
00:19:57,830 --> 00:20:02,654
矩阵将是你的西格玛V转置，但
matrix will be u Sigma V transpose but

318
00:20:02,659 --> 00:20:07,784
你知道这就像凝聚了一样
you know this is the like the condensed

319
00:20:07,789 --> 00:20:10,484
我扔掉东西的东西
thing where I've thrown away stuff

320
00:20:10,489 --> 00:20:13,394
这是自动为零，因为如果
that's automatically zero because if the

321
00:20:13,399 --> 00:20:15,584
排名只是我们喜欢的排名
rank is only our like if the rank was

322
00:20:15,589 --> 00:20:18,884
一个人认为排名不是我那个人
one suppose the rank wasn't one then I'd

323
00:20:18,889 --> 00:20:22,454
有一列时间一个西格玛时间
have one column times one sigma times

324
00:20:22,459 --> 00:20:27,014
一排，我可以做到这一点
one row right and I could do that count

325
00:20:27,019 --> 00:20:30,674
对于R等于1我现在有了我们的专栏
for R equal one now I have our columns

326
00:20:30,679 --> 00:20:38,924
所以这是M是否有西格玛是它的
so this is M by are there Sigma is its

327
00:20:38,929 --> 00:20:41,264
对角线当然所以我会得到我们的
diagonal of course so I'm gonna get our

328
00:20:41,269 --> 00:20:43,664
数字，而现在这个数字
numbers out of that and this one is now

329
00:20:43,669 --> 00:20:46,604
换句话说R也许我应该
R by n in other words maybe I should

330
00:20:46,609 --> 00:20:52,034
像在这里保存这一点，这是
like save this little bit here that was

331
00:20:52,039 --> 00:20:52,604
有帮助
helpful

332
00:20:52,609 --> 00:20:57,524
但现在我已经将M减少到R所以我
but now I've got M is reduced to R so I

333
00:20:57,529 --> 00:21:00,464
相信如果算上这三个我就会
believe that if I count these three I'll

334
00:21:00,469 --> 00:21:02,924
为a获取正确数量的参数
get the right number of parameters for a

335
00:21:02,929 --> 00:21:06,274
排名R矩阵，这不是那么明显
rank R matrix and that's not so obvious

336
00:21:06,279 --> 00:21:11,324
因为等级R矩阵不是我们的
because the rank R matrices are not a we

337
00:21:11,329 --> 00:21:14,924
如果我有一个等级R，则没有子空间
don't have a subspace if I had a rank R

338
00:21:14,929 --> 00:21:16,935
矩阵到另一个秩R矩阵
matrix to another rank R matrix

339
00:21:16,940 --> 00:21:19,784
这个等级可能大到2r和
well the rank could be as big as 2r and

340
00:21:19,789 --> 00:21:25,844
可能会这样，所以我们你知道它
probably will be so so we you know it's

341
00:21:25,849 --> 00:21:28,394
得到你的一点点有趣
just a little interesting to get your

342
00:21:28,399 --> 00:21:32,354
交给R级矩阵因为
hands on matrices of rank R because

343
00:21:32,359 --> 00:21:38,144
他们有点像数学的罪
they're kind of a sin like a well math

344
00:21:38,149 --> 00:21:40,634
人会称之为歧管一些
person would call it a manifold some

345
00:21:40,639 --> 00:21:42,405
一种表面
kind of a surface

346
00:21:42,410 --> 00:21:45,555
在矩阵空间中，你有没有
in matrix space yeah have you ever

347
00:21:45,560 --> 00:21:47,685
考虑矩阵空间，这是一个
thought about matrix space so that's a

348
00:21:47,690 --> 00:21:50,385
向量空间，因为我们可以添加矩阵
vector space because we can add matrices

349
00:21:50,390 --> 00:21:53,085
我们可以将它们乘以常数
we can multiply them by constants we can

350
00:21:53,090 --> 00:21:54,855
采取NIR组合所以他们是
take the NIR combination so they're

351
00:21:54,860 --> 00:21:57,255
如果我们，我们可以称它们为向量
there we could call them vectors if we

352
00:21:57,260 --> 00:22:01,305
这样就是M的向量空间
like that would be a vector space of M

353
00:22:01,310 --> 00:22:03,345
通过n矩阵将是什么
by n matrices what would be the

354
00:22:03,350 --> 00:22:07,425
那个空间支撑的尺寸所以
dimension of that space support so the

355
00:22:07,430 --> 00:22:09,315
矢量空间的所有三个四
vector space of all three by four

356
00:22:09,320 --> 00:22:15,635
具有什么维度的矩阵
matrices that that has what dimension

357
00:22:15,640 --> 00:22:18,165
1212因为你有12个号码
1212 because you've got 12 numbers to

358
00:22:18,170 --> 00:22:20,955
选择它，因为你是一个空间
choose and it is a space because you're

359
00:22:20,960 --> 00:22:24,195
如果我说3乘4矩阵，现在就加
gonna add now if I say 3 by 4 matrices

360
00:22:24,200 --> 00:22:29,975
等级2我再也没有空间了
of Rank 2 I don't have a space anymore

361
00:22:29,980 --> 00:22:33,705
那个词空间就是睡着了
that word spaces is is priscilly asleep

362
00:22:33,710 --> 00:22:37,425
保留用于意义向量空间
reserved for meaning vector space

363
00:22:37,430 --> 00:22:39,735
意思是我可以采取组合，但如果我
meaning I can take combinations but if I

364
00:22:39,740 --> 00:22:41,775
取2级矩阵加2级
take a rank 2 matrix plus a rank 2

365
00:22:41,780 --> 00:22:44,895
矩阵我不是那么它是一种表面
matrix I'm not so it's sort of a surface

366
00:22:44,900 --> 00:22:50,745
在12D到3x4矩阵内
within 12 D the to the 3 by 4 matrices

367
00:22:50,750 --> 00:22:53,105
等级2的和我们即将找到
of Rank 2 and we're about to find the

368
00:22:53,110 --> 00:22:56,715
那个表面的尺寸会影响你的想法
dimension of that surface does your mind

369
00:22:56,720 --> 00:22:59,385
在12中可视化表面
sort of visualize the surface in 12

370
00:22:59,390 --> 00:23:03,165
尺寸是的，好好试一试
dimensions yeah well give it a shot

371
00:23:03,170 --> 00:23:07,425
无论如何，但那表面可以
anyway and but what that surface could

372
00:23:07,430 --> 00:23:11,265
有11维可以这么说
have the 11 dimensional so to speak like

373
00:23:11,270 --> 00:23:16,245
在本地意味着它不必
meaning locally the it wouldn't have to

374
00:23:16,250 --> 00:23:20,175
是一个计划的11维飞机
be a plan 11 dimensional plane going

375
00:23:20,180 --> 00:23:23,055
通过其实起源它不会
through the origin in fact it wouldn't

376
00:23:23,060 --> 00:23:24,855
经过由来是因为产地
go through the origin because the origin

377
00:23:24,860 --> 00:23:27,645
没有排名是如此，所以它是某种
won't have rank are so so it's some kind

378
00:23:27,650 --> 00:23:29,895
一个表面，也许它有一些
of a surface and maybe it's got some

379
00:23:29,900 --> 00:23:33,585
不同的部分可能有些聪明
different pieces probably some smart

380
00:23:33,590 --> 00:23:35,655
人知道那个表面看起来像什么
person knows what that surface looks

381
00:23:35,660 --> 00:23:38,565
等，但是我们只是要找出
like but we're just going to find out

382
00:23:38,570 --> 00:23:41,985
有些人对它的数量有所了解
some sense something about its number of

383
00:23:41,990 --> 00:23:44,355
参数作为局部维度我
parameters as local dimension well I

384
00:23:44,360 --> 00:23:48,465
知道这个答案是我所得到的
know that this answer is R cuz I've got

385
00:23:48,470 --> 00:23:52,425
我们的Sigma和这个我非常好
our Sigma's and this one I'm pretty good

386
00:23:52,430 --> 00:23:55,695
但现在却不是
at but now it's not

387
00:23:55,700 --> 00:24:00,435
它是Rbyn所以它不是在这里
it's R by n so it's instead of here here

388
00:24:00,440 --> 00:24:04,784
我们是M但现在在这里R是R所以我
our was M but now down here R is R so I

389
00:24:04,789 --> 00:24:13,475
认为它是n减1/2
think it's are n minus 1/2

390
00:24:13,480 --> 00:24:17,475
什么是M所以现在它是一个
what's that is that an M so now it's an

391
00:24:17,480 --> 00:24:25,995
RR加1我想我想和什么
R R plus 1 I think I think and what

392
00:24:26,000 --> 00:24:31,294
关于你，所以U将会是相似的
about the u so U is going to be similar

393
00:24:31,299 --> 00:24:34,424
除了在这里，我们已经到了
except instead of the end here we've got

394
00:24:34,429 --> 00:24:38,955
一个MI所以我想我们会为你做的
an M I so I think we for you will will

395
00:24:38,960 --> 00:24:43,695
有M减1加M减2加等等
have M minus 1 plus M minus 2 plus so so

396
00:24:43,700 --> 00:24:47,534
让我在这里写下M减去1所以你
let me write it here M minus 1 so you

397
00:24:47,539 --> 00:24:49,755
我在这里谈论你，这是我们的
I'm talking about you here it's got our

398
00:24:49,760 --> 00:24:53,085
列第一个有M减1
columns the first one has M minus 1

399
00:24:53,090 --> 00:24:55,245
因为我扔了一个，因为它是一个
because I throw away one because it's a

400
00:24:55,250 --> 00:25:00,554
单位向量最多M减去R我们的列
unit vector up to M minus R ours column

401
00:25:00,559 --> 00:25:05,264
好的，现在怎么样那是什么
okay and now so how what what is that

402
00:25:05,269 --> 00:25:07,245
加起来我把所有的M都放好了
add up to well I put all the M's

403
00:25:07,250 --> 00:25:11,625
因此，我们的M让我说先生
together so that's our M let me say mr

404
00:25:11,630 --> 00:25:15,404
然后我中减去1加2
and then I'm subtracting off 1 plus 2

405
00:25:15,409 --> 00:25:18,644
加3到R现在再告诉我什么
Plus 3 up to R now tell me again what

406
00:25:18,649 --> 00:25:21,735
这加起来一加二加三
that adds up to one plus two plus three

407
00:25:21,740 --> 00:25:25,955
停在我们这就是我们在这里所拥有的
stop at our that's what we had here and

408
00:25:25,960 --> 00:25:29,235
我们已经把它用于V而且我们已经得到了它
we've got it for V and we've got it

409
00:25:29,240 --> 00:25:36,105
再来这里减去1/2我们的R加1你
again here minus 1/2 our R plus 1 you

410
00:25:36,110 --> 00:25:36,884
处之泰然
okay with that

411
00:25:36,889 --> 00:25:41,534
现在我只是想把它们加起来所以我
and now I just want to add them up so I

412
00:25:41,539 --> 00:25:45,825
有MR和我已经R然后我有
have M R and I've n R and then I have

413
00:25:45,830 --> 00:25:48,735
其中两个让我让我得到它
two of these so let me let me get it

414
00:25:48,740 --> 00:25:54,014
MR和n现在我必须这样做
here M R and n are and now I have to

415
00:25:54,019 --> 00:25:58,274
看看如此先检查我现在检查了
look at so mr check n are checked now I

416
00:25:58,279 --> 00:26:00,855
有两个这样的家伙，所以他们结合起来
have two of these guys so they combine

417
00:26:00,860 --> 00:26:05,294
进入R平方加R然后IR
into R squared plus R and then I R

418
00:26:05,299 --> 00:26:07,325
平方
squared

419
00:26:07,330 --> 00:26:09,585
是的，减去R平方
yeah minus R squared

420
00:26:09,590 --> 00:26:12,615
抱歉，他们合并为负R平方
sorry they combined into minus R squared

421
00:26:12,620 --> 00:26:15,765
加上R然后这里有一个进来
plus R and then here's a are coming in

422
00:26:15,770 --> 00:26:18,075
加上我认为我们有一个减去R
with a plus I think we have a minus R

423
00:26:18,080 --> 00:26:23,435
平方，这是正确的答案
squared and that is the right answer

424
00:26:23,440 --> 00:26:28,995
是的好了，所以我花了一点时间比我
yep okay so I took a bit longer than I

425
00:26:29,000 --> 00:26:31,925
意，但它这是一个数字，
intended but it this is a number that

426
00:26:31,930 --> 00:26:34,185
这很有趣
that's sort of interesting

427
00:26:34,190 --> 00:26:36,675
我提到马鞍点有点像
I mentioned saddle points sort of like

428
00:26:36,680 --> 00:26:39,675
分开最大值和最小值只
separately from maxima and minima just

429
00:26:39,680 --> 00:26:42,855
因为它们绝对不那么容易
because they are definitely not as easy

430
00:26:42,860 --> 00:26:45,795
和你一起工作明白我的意思
to work with you understand what I mean

431
00:26:45,800 --> 00:26:51,245
通过涉及马鞍点矩阵
by saddle points matrices involved have

432
00:26:51,250 --> 00:26:55,485
这些都不会肯定
are not positive definite those would go

433
00:26:55,490 --> 00:26:58,335
最大限度，他们不是负面的
with a maximum they're not negative

434
00:26:58,340 --> 00:27:02,355
他们不是消极的，他们并不好
they're not negative they're not well

435
00:27:02,360 --> 00:27:04,365
那些将与最大值和最小值
those would go with maxima and minima

436
00:27:04,370 --> 00:27:07,005
但我们是我们之间，在寻找
but we're we're looking in between so

437
00:27:07,010 --> 00:27:10,245
马鞍点好吧我会开始的
saddle points okay well I'll get going

438
00:27:10,250 --> 00:27:13,185
对那些没关系，我有点意识到这一点
on those okay I I sort of realized that

439
00:27:13,190 --> 00:27:16,995
马鞍有两个主要来源
there are two main sources of saddle

440
00:27:17,000 --> 00:27:22,125
他们中的一个点是当我有
points one of them is when I have

441
00:27:22,130 --> 00:27:25,905
当我说我的时候出现的问题
problems that when I let's say I

442
00:27:25,910 --> 00:27:30,665
最小化，这将是约束
minimize so this will be the constraint

443
00:27:30,670 --> 00:27:32,655
马鞍点来自于
saddle points have come from the

444
00:27:32,660 --> 00:27:37,005
约束让勒布朗拉格朗日会如此
constraint so LeBron Lagrange's gonna be

445
00:27:37,010 --> 00:27:39,555
对这些马鞍点负责
responsible for these saddle points so

446
00:27:39,560 --> 00:27:41,805
我们可能有像一些极小问题
we might have some minimum problem like

447
00:27:41,810 --> 00:27:45,965
尽量减少一些肯定的事情
minimize some positive definite thing

448
00:27:45,970 --> 00:27:48,885
当然，如果我们不再说了
and of course if we don't say anymore

449
00:27:48,890 --> 00:27:52,025
最小值是零正确的，因为
the minimum is zero right because

450
00:27:52,030 --> 00:27:55,725
否则它是积极的，但我们要走了
otherwise it's positive but we're going

451
00:27:55,730 --> 00:27:59,985
把约束斧等于B所以这个
to put on constraints ax equal B so this

452
00:27:59,990 --> 00:28:04,955
是经典的约束
is the classical constrained

453
00:28:04,960 --> 00:28:09,495
优化问题二次成本
optimization problem quadratic cost

454
00:28:09,500 --> 00:28:13,095
函数线性约束我们可以
function linear constraints we could

455
00:28:13,100 --> 00:28:17,115
完全解决这个问题，但让我们看看
solve this exactly but let's just see

456
00:28:17,120 --> 00:28:20,825
马鞍点会上升的地方
where saddle point is going to rise so

457
00:28:20,830 --> 00:28:22,654
所以有
so there's

458
00:28:22,659 --> 00:28:26,315
是肯定的，但现在我们如何
is positive-definite but now how do we

459
00:28:26,320 --> 00:28:28,955
妥善处理这个问题
deal with that problem well the ground

460
00:28:28,960 --> 00:28:33,364
说做什么Lagaan说看看
said what to do Lagaan said look at the

461
00:28:33,369 --> 00:28:40,744
拉格朗日好吧，他介绍了lambda
Lagrangian okay he introduced lambda so

462
00:28:40,749 --> 00:28:45,154
这个X在这个X中是n维
this X is in this X is in n dimensions

463
00:28:45,159 --> 00:28:47,794
这是一个N乘n矩阵，但我有M.
that's an N by n matrix but I have M

464
00:28:47,799 --> 00:28:55,435
约束因此矩阵a是M乘以n
constraints so the matrix a is M by n

465
00:28:55,440 --> 00:28:59,794
我有M个约束然后我要去
I've M constraints and then I'm going to

466
00:28:59,799 --> 00:29:02,104
按照规则介绍Mlagron
follow the rules and introduce M lagron

467
00:29:02,109 --> 00:29:06,114
小应用者
smalt appliers

468
00:29:06,119 --> 00:29:13,174
这是一个M，然后是整洁的部分
that's an M and then the neat part of

469
00:29:13,179 --> 00:29:15,874
呻吟，什么是这个好
the groan and what what is this well

470
00:29:15,879 --> 00:29:21,024
这是我参加的功能，然后我
it's I take the function and then I

471
00:29:21,029 --> 00:29:23,884
介绍记住lambda是一个向量
introduce remember lambda is a vector

472
00:29:23,889 --> 00:29:25,955
现在不只是一个数字
now not just a number

473
00:29:25,960 --> 00:29:27,724
我们有一些应用，其中它是
we had some application where it was

474
00:29:27,729 --> 00:29:30,004
只是，有一只约束，但是
just there was just one constraint but

475
00:29:30,009 --> 00:29:32,435
现在我有M个约束，所以我拿了一个
now I have M constraints so I take a

476
00:29:32,440 --> 00:29:36,274
lambdasolambdatransposetimesax
lambda so lambda transpose times ax

477
00:29:36,279 --> 00:29:41,254
减去B和加号或减号
minus B and the plus or the minus sign

478
00:29:41,259 --> 00:29:45,215
这里并不重要我的意思是你可以
here is not important I mean you can

479
00:29:45,220 --> 00:29:47,044
选择它，因为那将决定
choose it because that'll determine the

480
00:29:47,049 --> 00:29:50,315
lambda的标志，但不管怎样
sign of lambda but either way it's

481
00:29:50,320 --> 00:29:51,575
正确
correct

482
00:29:51,580 --> 00:29:54,514
好的，所以我们已经进入了它
okay so we've entered it we've

483
00:29:54,519 --> 00:29:58,744
介绍了一个现在依赖的功能
introduced a function that now depends

484
00:29:58,749 --> 00:30:03,274
X和还对Lambda和和有
on X and also on lambda and and there's

485
00:30:03,279 --> 00:30:08,344
a和他们互相繁衍和我的
a and they multiply each other in and my

486
00:30:08,349 --> 00:30:12,695
重点是拉格龙说的
point is that lagron says take the

487
00:30:12,700 --> 00:30:15,994
关于X和λ的衍生物
derivatives with respect to X and lambda

488
00:30:15,999 --> 00:30:18,394
所以这是他很酷的事情
so that's the cool thing that he's

489
00:30:18,399 --> 00:30:21,004
他说，如果你只是创造
contributed he says if you only create

490
00:30:21,009 --> 00:30:24,484
我的功能现在你可以拿X.
my function now you can take X

491
00:30:24,489 --> 00:30:26,705
衍生物和λ衍生物
derivative and lambda derivative that

492
00:30:26,710 --> 00:30:30,754
会给你n个X的方程式
will give you n equations for the X's

493
00:30:30,759 --> 00:30:33,244
从这一个来自X衍生物和
from this one from the X derivative and

494
00:30:33,249 --> 00:30:35,415
M方程从lambda到
M equations from the lambda to

495
00:30:35,420 --> 00:30:38,415
它将是n加M它将决定
it'll be n plus M it'll determine the

496
00:30:38,420 --> 00:30:41,505
好的X和lambda但是我说的
good X and the lambda but I'm saying it

497
00:30:41,510 --> 00:30:44,295
这一切都是真的，但都很重要
that's all true and all important but

498
00:30:44,300 --> 00:30:46,755
我是说X和那对X.
I'm saying that the X and that pair X

499
00:30:46,760 --> 00:30:49,875
lambda将成为这个问题的一个重点
lambda will be a saddle point of this

500
00:30:49,880 --> 00:30:52,635
功能此功能有鞍点
function this function has saddle points

501
00:30:52,640 --> 00:30:56,925
不是没有最大的好
not not a maximum okay

502
00:30:56,930 --> 00:30:59,115
让我们看看衍生品吧
let's just take the derivatives and see

503
00:30:59,120 --> 00:31:02,475
我们得到这样与衍生品
what we get so the derivatives with

504
00:31:02,480 --> 00:31:06,375
通过DX尊重XD我现在是一个
respect to X D by DX I'm ex is now a

505
00:31:06,380 --> 00:31:09,185
矢量所以我真的应该说
vector so I really should say the

506
00:31:09,190 --> 00:31:13,625
我得到了X方向的渐变
gradient in the X direction I get I get

507
00:31:13,630 --> 00:31:18,885
SX和这里是规范的衍生物
SX and here the derivative of the spec

508
00:31:18,890 --> 00:31:23,985
2x这就是转置
2x what that's that is a transpose

509
00:31:23,990 --> 00:31:26,955
lambda因为这是点积
lambda because this is the dot product

510
00:31:26,960 --> 00:31:29,805
一个转置lambda与X你，你
of a transpose lambda with X you you

511
00:31:29,810 --> 00:31:32,045
我知道我已经把括号括起来了
know I've put parentheses around it and

512
00:31:32,050 --> 00:31:35,025
遵循转置规则，这样
followed the transpose rule so that's

513
00:31:35,030 --> 00:31:37,335
转置lambda的点积
the dot product of a transpose lambda

514
00:31:37,340 --> 00:31:39,675
使用X，它在X中是线性的，所以它是
with X it's linear in X so it's

515
00:31:39,680 --> 00:31:41,685
衍生物只是一个转置lambda
derivative is just a transpose lambda

516
00:31:41,690 --> 00:31:47,325
那就是现在我拿另一个
and that's and now I take the other one

517
00:31:47,330 --> 00:31:52,395
lambda衍生物lambda
the the lambda derivative the lambda

518
00:31:52,400 --> 00:31:54,195
衍生物这不依赖于lambda
derivative this doesn't depend on lambda

519
00:31:54,200 --> 00:31:56,835
lambda导数只是X减去
the lambda derivative is just a X minus

520
00:31:56,840 --> 00:32:03,525
B它带来了约束
B it brings back the constraints so

521
00:32:03,530 --> 00:32:06,735
这很简单，不，甚至没有
that's pretty simple no it doesn't even

522
00:32:06,740 --> 00:32:08,805
需要很多思考，因为你只是
require much thought because you just

523
00:32:08,810 --> 00:32:10,875
知道约束回来，
know the constraints are coming back and

524
00:32:10,880 --> 00:32:14,085
当然应该把B放在这上面
of course B should be put over on this

525
00:32:14,090 --> 00:32:18,855
因为它是一个常数所以我们
side because it's a constant so there we

526
00:32:18,860 --> 00:32:25,395
看到一个块我们看到一个非常重要的
see to a block we see a important very

527
00:32:25,400 --> 00:32:28,095
重要的一类问题和问题
important class of problems and the

528
00:32:28,100 --> 00:32:30,105
矩阵我们看到我们可以写这个
matrix we're seeing we could write this

529
00:32:30,110 --> 00:32:35,615
以块矩阵形式减去转置
in block matrix form s minus a transpose

530
00:32:35,620 --> 00:32:39,525
哦，我要把它改成加号
oh I'm going to change that to a plus

531
00:32:39,530 --> 00:32:43,235
因为我更加人还好的一个
because I'm more of a plus person okay a

532
00:32:43,240 --> 00:32:45,265
颠倒
transpose

533
00:32:45,270 --> 00:32:50,665
当我拿走时，是的，是的
and a yeah yeah when I when I took the

534
00:32:50,670 --> 00:32:51,955
导致这打破了我的lambda
derivative this break the lambda I

535
00:32:51,960 --> 00:32:54,355
我没有把减号放在这里
didn't put the minus sign in here and I

536
00:32:54,360 --> 00:32:57,115
我不想让它成为一个加号
didn't want to so let's make it a plus a

537
00:32:57,120 --> 00:33:02,785
再没有什么也没有和
and then there's no nothing there and

538
00:33:02,790 --> 00:33:07,165
X和lambda以及零和
the X and the lambda and the zero and

539
00:33:07,170 --> 00:33:15,255
B是受约束的模型
the B that is the model of a constrained

540
00:33:15,260 --> 00:33:17,515
最小的问题
minimum a minimum problem with

541
00:33:17,520 --> 00:33:20,875
约束它是一个模型，因为
constraints it's a model because the

542
00:33:20,880 --> 00:33:23,125
这里的函数是二次的和
function here is quadratic and the

543
00:33:23,130 --> 00:33:25,735
约束是线性的，你知道
constraints are linear it's a you know

544
00:33:25,740 --> 00:33:30,655
它在六个过程中无处不在
it's in in course six it's everywhere

545
00:33:30,660 --> 00:33:34,165
不断地不断出现
constant constantly appearing as the

546
00:33:34,170 --> 00:33:37,435
最简单的模型好，今天我的观点
simplest model okay and my point today

547
00:33:37,440 --> 00:33:43,605
只是解决方案Xlambda那个
is just that solution X lambda that that

548
00:33:43,610 --> 00:33:48,955
总共解决了X的问题
that total solution the the X together

549
00:33:48,960 --> 00:33:51,925
有一个lambda，这是一个马鞍
with a lambda that that is a saddle

550
00:33:51,930 --> 00:33:58,285
拉格朗日函数的重点是它
point of the lagrangian function l it's

551
00:33:58,290 --> 00:34:01,195
一个马鞍点不是最小的它是那种
a saddle point not a minimum it's sort

552
00:34:01,200 --> 00:34:06,025
在X方向上最小的因为
of a minimum in the X direction because

553
00:34:06,030 --> 00:34:08,605
这是肯定的
this is positive definite that makes

554
00:34:08,610 --> 00:34:11,515
这使得它成为X的函数
that makes that as a function of X it's

555
00:34:11,520 --> 00:34:18,295
上升，但不知何故的外观
going up but somehow the appearance of

556
00:34:18,300 --> 00:34:23,005
lambda使这个矩阵不确定
lambda makes this matrix indefinite it

557
00:34:23,010 --> 00:34:26,245
开始肯定但它有这个
starts positive definite but it has this

558
00:34:26,250 --> 00:34:28,615
转置a和零
a transpose a and that zero

559
00:34:28,620 --> 00:34:31,315
它不可能是大力水手看那个
it couldn't be Popeye look at that

560
00:34:31,320 --> 00:34:33,654
矩阵我看到它不是肯定的
matrix I see it's not positive definite

561
00:34:33,659 --> 00:34:36,295
我怎么看他为什么这么说呢
what do I see him why do I say that

562
00:34:36,300 --> 00:34:38,725
我立刻看着那个矩阵
immediately when I look at that matrix

563
00:34:38,730 --> 00:34:40,575
它不是一个正定矩阵
it's not a positive definite matrix

564
00:34:40,580 --> 00:34:46,885
因为当我看到零时
because when I see that zero on the

565
00:34:46,890 --> 00:34:50,215
对角线肯定是正面的
diagonal that shoots positive definitely

566
00:34:50,220 --> 00:34:52,885
不能采取采取作为示例S
couldn't be take take as an example s

567
00:34:52,890 --> 00:34:57,805
等于31和10取该矩阵
equal three 1 and 1 0 take that matrix

568
00:34:57,810 --> 00:35:02,755
随便闻到我买它2
just random smell I made it to buy 2

569
00:35:02,760 --> 00:35:06,985
而不是大小n加上你知道如何
instead of size n plus in do you see how

570
00:35:06,990 --> 00:35:09,025
我知道那个特征值
do I know that the eigenvalues of that

571
00:35:09,030 --> 00:35:14,905
矩阵一是加号，一个是减号
matrix one is plus and one is minus the

572
00:35:14,910 --> 00:35:20,695
决定因素是否定的，这告诉我
determinant is negative so that tells me

573
00:35:20,700 --> 00:35:22,735
马上，一个是加号，一个是-
right away that one is plus and one is -

574
00:35:22,740 --> 00:35:25,705
谢谢是的，决定因素是消极的
thanks yes yes determinant is negative

575
00:35:25,710 --> 00:35:27,955
不知怎的，这里的决定因素是
and somehow here the determinant is

576
00:35:27,960 --> 00:35:30,465
类似的计算会产生一个
similar calculation would produce a

577
00:35:30,470 --> 00:35:34,705
使用-cuz转置a或某事
transpose a or something with the - cuz

578
00:35:34,710 --> 00:35:38,125
我会这样好，我能我能
I'm going this way well I could I could

579
00:35:38,130 --> 00:35:42,835
做得比这更好，但你看到了
do better than that but you saw the

580
00:35:42,840 --> 00:35:46,315
指出那个简单的矩阵
point that that matrix that simple

581
00:35:46,320 --> 00:35:52,405
这个例子就是特征值
example of this is has eigenvalues are

582
00:35:52,410 --> 00:35:55,405
这两个星座让我快说和
both signs let me just quickly say and

583
00:35:55,410 --> 00:35:58,945
我会把它放在笔记中或那里
I'll put it in the notes or in that

584
00:35:58,950 --> 00:36:02,655
章节我猜这一切都将到来
chapter I guess it all this is coming is

585
00:36:02,660 --> 00:36:09,505
仍然是3.2，原来是4.2
still 3.2 the that was originally 4.2

586
00:36:09,510 --> 00:36:14,005
并且你会看到它，所以我想要什么
and and you'll see it so what do I want

587
00:36:14,010 --> 00:36:16,915
说我想说那个例子
to say I'd like to say that that example

588
00:36:16,920 --> 00:36:19,975
我非常有说服力
is pretty convincing to me that that

589
00:36:19,980 --> 00:36:26,905
如果你说话，这些KKT矩阵
these KKT matrices if if you talk to

590
00:36:26,910 --> 00:36:36,445
人们在优化中是kirusha
people in in optimization that's kirusha

591
00:36:36,450 --> 00:36:39,175
kuhn和tucker三个着名的家伙和
kuhn and tucker three famous guys and

592
00:36:39,180 --> 00:36:43,285
这些是KKT的条件
these are the KKT conditions that that

593
00:36:43,290 --> 00:36:47,155
他们得出了以下拉格罗斯的权利
they derived following lagrosse right

594
00:36:47,160 --> 00:36:50,275
我的意思是，是的，这是一个
and my point is yeah and this is a

595
00:36:50,280 --> 00:36:54,595
典型的排序，所以这是一个不确定的
typical sort so it's it's an indefinite

596
00:36:54,600 --> 00:37:01,375
矩阵我相信如果我这样做的话
matrix I believe it has that one if I do

597
00:37:01,380 --> 00:37:04,855
消除是的告诉我这是一个
elimination yeah tell me this this is a

598
00:37:04,860 --> 00:37:07,585
很好的方式来看待它我想
good way to look at it suppose I do

599
00:37:07,590 --> 00:37:10,525
消除这个或这个
elimination on on this one or on this

600
00:37:10,530 --> 00:37:11,485
一口井假设
one well suppose

601
00:37:11,490 --> 00:37:13,915
我在那里消灭的是第一个
I do elimination there what is the first

602
00:37:13,920 --> 00:37:19,135
枢纽三正，所以现在让我转
pivot three positive so now let me turn

603
00:37:19,140 --> 00:37:21,805
到这里如果我做消除怎么办？
down to here what if I do elimination on

604
00:37:21,810 --> 00:37:27,114
这个块矩阵然后我在这里启动
this block matrix then I start up here

605
00:37:27,119 --> 00:37:30,955
而我和第一个支点是积极的
and I and that first pivot is positive

606
00:37:30,960 --> 00:37:33,535
再一次，这是s是积极的
again right this is s is a positive

607
00:37:33,540 --> 00:37:35,245
确定矩阵不要忘记
definite matrix don't forget

608
00:37:35,250 --> 00:37:38,725
实际上，前n个枢纽都将是
in fact the first n pivots will all be

609
00:37:38,730 --> 00:37:40,825
积极因为第一和枢轴
positive because the first and pivot

610
00:37:40,830 --> 00:37:42,715
搜索你正在努力解决这个问题
search you're working away in this

611
00:37:42,720 --> 00:37:45,265
角落，如果你只是如果你只是
corner and if you only if you're only

612
00:37:45,270 --> 00:37:47,845
想着这个角落的第一个角落
thinking about the first n this corner

613
00:37:47,850 --> 00:37:51,745
是n的大小n然后你甚至没有
is size n by n then you you don't even

614
00:37:51,750 --> 00:37:54,955
看到你正在做一些减法和
see a you're doing some subtractions and

615
00:37:54,960 --> 00:37:58,075
除了枢纽本身，我会做那些
I'll do those but the pivots themselves

616
00:37:58,080 --> 00:38:00,685
来自s的所有人都来了，S是
are coming all coming from s and S is

617
00:38:00,690 --> 00:38:03,085
肯定的，所以我们知道其中一个
positive definite so we know that one of

618
00:38:03,090 --> 00:38:04,975
对正定矩阵的检验
the tests for a positive definite matrix

619
00:38:04,980 --> 00:38:08,095
所有的支点都是积极的，所以我想
is all pivots are positive so I think

620
00:38:08,100 --> 00:38:10,855
所有n个第一支点都将是
all n of the first pivots will be

621
00:38:10,860 --> 00:38:14,185
积极的，当我们使用它们时，让我们
positive and and when we use them let's

622
00:38:14,190 --> 00:38:16,445
刚刚看到，当我们使用它们会发生什么
just see what happens when we use them

623
00:38:16,450 --> 00:38:17,655
[掌声]
[Applause]

624
00:38:17,660 --> 00:38:22,255
所以这就是我开始的KKT矩阵
so so here's the KKT matrix that I start

625
00:38:22,260 --> 00:38:30,795
我最终得到了什么？
with and what do I end up with well

626
00:38:30,800 --> 00:38:33,805
我真正在做的是我在成倍增长
really what I'm doing is I'm multiplying

627
00:38:33,810 --> 00:38:37,315
通过诸如此类的话块行
that block row by something to and

628
00:38:37,320 --> 00:38:40,795
减去杀死我就是这样的
subtracting to kill that a I'm so the

629
00:38:40,800 --> 00:38:44,395
这些排很近，让我放手
these rows well near enough let me let

630
00:38:44,400 --> 00:38:46,975
我做阻止消除块
me do block elimination block

631
00:38:46,980 --> 00:38:49,045
消除就像是容易我没有
elimination is like easier I don't have

632
00:38:49,050 --> 00:38:52,915
写下所有的小数字
to write down all little tiny numbers so

633
00:38:52,920 --> 00:38:55,245
我只想将这一行乘以
I just want to multiply this row by

634
00:38:55,250 --> 00:39:00,025
东西告诉我什么和减去
something tell me what and subtract from

635
00:39:00,030 --> 00:39:03,025
此第二行只是我想有
this second row except I suppose there

636
00:39:03,030 --> 00:39:07,495
我猜他们是数字或字母
are numbers or letters I guess they are

637
00:39:07,500 --> 00:39:10,945
字母我先把它乘以什么
letters what do I multiply that first

638
00:39:10,950 --> 00:39:17,915
逐行减去
row by and subtract

639
00:39:17,920 --> 00:39:21,545
让我们看看这些是不是很小
let's see if these were just little tiny

640
00:39:21,550 --> 00:39:27,515
数字就像3110我是什么
numbers as like in 3 1 1 0 what do I

641
00:39:27,520 --> 00:39:29,555
将该行乘以并减去
multiply that row by and subtract from

642
00:39:29,560 --> 00:39:34,295
这个我乘以一个/s吧
this I multiply by a / s right I do

643
00:39:34,300 --> 00:39:36,485
乘以这使一个AA/S
multiplied by a / s which puts an A

644
00:39:36,490 --> 00:39:39,215
然后我在这里减去所以我会
there then I subtract so here I'll

645
00:39:39,220 --> 00:39:41,855
通过A/S繁殖，但我这些都是
multiply by a / s but I these are

646
00:39:41,860 --> 00:39:49,145
矩阵，所以当我乘以s乘以a
matrices so while I multiply by s by a

647
00:39:49,150 --> 00:39:53,525
SN我乘以倒数
SN I'm multiplying by a s inverse times

648
00:39:53,530 --> 00:39:56,405
我得到了一个，然后我减去了我
this s I get a and then I subtract and I

649
00:39:56,410 --> 00:39:59,675
得到0，当我乘以这个家伙
get to 0 and when I multiply by this guy

650
00:39:59,680 --> 00:40:02,975
减去因为我是减去的我
and subtract I get minus because I'm

651
00:40:02,980 --> 00:40:08,995
减去这个东西-作为逆a
subtracting this thing - a s inverse a

652
00:40:09,000 --> 00:40:16,235
转置是块消除
transpose that was block elimination

653
00:40:16,240 --> 00:40:21,835
换句话说，这只是我
which just in other words it's just I

654
00:40:21,840 --> 00:40:25,745
你已经了解了2个2矩阵
you've learned about 2 by 2 matrices you

655
00:40:25,750 --> 00:40:29,435
现在知道3x加4y等于7和东西
know 3x plus 4y equals 7 and stuff now

656
00:40:29,440 --> 00:40:32,345
我只是用块来做这件事
I'm just doing it with with with blocks

657
00:40:32,350 --> 00:40:35,125
而不是单个数字，但你看
instead of single numbers but you see

658
00:40:35,130 --> 00:40:38,825
这产生了那些积极的支点和
this produced those positive pivots and

659
00:40:38,830 --> 00:40:41,935
你能告诉我关于那个矩阵的什么
what can you tell me about that matrix

660
00:40:41,940 --> 00:40:45,245
什么样的什么可以告诉我
what what kind of what can you tell me

661
00:40:45,250 --> 00:40:47,765
关于标志或特征值或
about the signs or the eigenvalues or

662
00:40:47,770 --> 00:40:54,725
不管这个矩阵是什么意思
whatever of this matrix suppose this was

663
00:40:54,730 --> 00:40:57,365
什么你能告诉我的身份
the identity what could you tell me

664
00:40:57,370 --> 00:41:00,925
关于-aatranspose
about - a a transpose

665
00:41:00,930 --> 00:41:03,725
-aa转换我的声音应该
- a a transpose a my voice should

666
00:41:03,730 --> 00:41:07,805
强调-那就是那个矩阵
emphasize that - it's that matrix there

667
00:41:07,810 --> 00:41:13,085
是负面的，所以虽然
is negative definite so although the

668
00:41:13,090 --> 00:41:15,695
来自的下一组mm枢轴
next set of mm pivots that come from

669
00:41:15,700 --> 00:41:18,805
在这里，我们都会消极，所以我得到了M.
here we'll all be negative so I get M

670
00:41:18,810 --> 00:41:29,794
而nn++n-枢轴
rather n n+ + n- pivots

671
00:41:29,799 --> 00:41:34,215
然后我记得那个枢轴
and and then I remember that the pivots

672
00:41:34,220 --> 00:41:36,015
居然有相同的符号
actually have the same sign as the

673
00:41:36,020 --> 00:41:39,195
特征值，这只是一个美丽的事实
eigenvalues that's just a beautiful fact

674
00:41:39,200 --> 00:41:42,614
我们知道这是肯定的
we know that for for positive definite

675
00:41:42,619 --> 00:41:45,135
那些特征值都是正的
ones the eigenvalues are all positive

676
00:41:45,140 --> 00:41:47,445
枢轴都是积极的，但它是
the pivots are all positive but it's

677
00:41:47,450 --> 00:41:50,114
甚至比这更好的，如果我们有
even better than that that if we have

678
00:41:50,119 --> 00:41:53,505
一些混合物用于枢轴的迹象
some mixture for the signs of the pivots

679
00:41:53,510 --> 00:41:56,054
这告诉我们的迹象
that tells us the signs of the

680
00:41:56,059 --> 00:41:59,324
特征值，这是非常巧妙的事实
eigenvalues that's really neat fact so

681
00:41:59,329 --> 00:42:02,564
我只会写下标志加上和
I'll just write that down signs plus and

682
00:42:02,569 --> 00:42:09,645
减去枢轴的迹象给我们带来了好处
minus signs of pivots give us the plus

683
00:42:09,650 --> 00:42:15,455
或者我伽纳彻减号
or minus signs of the I ganache

684
00:42:15,460 --> 00:42:20,324
所以我潜入了一个漂亮的矩阵
so I've sneaked sneaked in a nice matrix

685
00:42:20,329 --> 00:42:23,655
偏爱对称的定理
theorem that prefer for symmetric

686
00:42:23,660 --> 00:42:29,584
矩阵这是对称矩阵没关系
matrices this is symmetric matrices okay

687
00:42:29,589 --> 00:42:34,445
这就是我想说的约
that's what I wanted to say about

688
00:42:34,450 --> 00:42:37,844
约束和马鞍点即将到来
constraints and saddle points coming

689
00:42:37,849 --> 00:42:41,025
从那里然后我想说
from there and then I want to say

690
00:42:41,030 --> 00:42:44,505
关于约束而不是约束
something about constraints and not

691
00:42:44,510 --> 00:42:46,755
约束现在我要看看
constraints now I'm going to look at the

692
00:42:46,760 --> 00:42:53,354
第二个马鞍点来源
second source of of saddle points so

693
00:42:53,359 --> 00:43:00,935
这些是这个的鞍座
these are the saddles from this

694
00:43:00,940 --> 00:43:14,025
我们所知道的非凡功能
remarkable function that we know so I

695
00:43:14,030 --> 00:43:18,885
现在有一个对称矩阵S可以
now have a symmetric matrix S could be

696
00:43:18,890 --> 00:43:20,834
通常也是肯定的
even positive definite usually it is

697
00:43:20,839 --> 00:43:25,064
你在这里知道R的名字是什么
here do you know what the name for R is

698
00:43:25,069 --> 00:43:28,275
它是一个比例或它命名的商
it's a ratio or a quotient it's named

699
00:43:28,280 --> 00:43:31,515
有人从我们的人开始
after somebody starting with our who's

700
00:43:31,520 --> 00:43:34,604
这真的是一个瑞利
that really right it's a Rayleigh

701
00:43:34,609 --> 00:43:39,314
商
quotient

702
00:43:39,319 --> 00:43:44,004
什么是最大的价值
and what is the largest value possible

703
00:43:44,009 --> 00:43:49,794
我们瑞利商的价值
value of the Rayleigh quotient we've

704
00:43:49,799 --> 00:43:54,804
看到这个想法是最大的价值
seen this idea it is the maximum value

705
00:43:54,809 --> 00:43:56,844
那个比率的瑞利商
of that Rayleigh quotient of that ratio

706
00:43:56,849 --> 00:44:02,754
是lambdamaxrightlambdaone
is is lambda max right lambda one the

707
00:44:02,759 --> 00:44:06,234
最大的一个，退出的是它
biggest one and the exit does it is the

708
00:44:06,239 --> 00:44:12,534
特征向量，因此max是lambda1
eigenvector right so max is is lambda 1

709
00:44:12,539 --> 00:44:21,385
并且在X等于Q，因为Q1转置s
and at X equal Q because Q 1 transpose s

710
00:44:21,390 --> 00:44:26,214
问1Q1转换Q1所以我是
Q 1 over Q 1 transpose Q 1 so I'm

711
00:44:26,219 --> 00:44:31,554
插入这个赢家和Q1是
plugging in this winner and s Q 1 is

712
00:44:31,559 --> 00:44:35,185
lambda1Q1对，这是第一个
lambda 1 Q 1 right it's the first

713
00:44:35,190 --> 00:44:38,754
特征向量，因此lambda1出来了
eigenvector and so a lambda 1 comes out

714
00:44:38,759 --> 00:44:44,004
所以我得到lambda1我知道一切
so I get lambda 1 I know everything

715
00:44:44,009 --> 00:44:48,054
关于那个和我知道的是，如果我把
about that and what I know is if I put

716
00:44:48,059 --> 00:44:52,764
在任何X，我知道如果我放入任何
in any X where do I know if I put in any

717
00:44:52,769 --> 00:44:56,274
X什么，看看这个数字是什么
X whatever and look at this number what

718
00:44:56,279 --> 00:45:00,324
我知道这个数字是否更小
do I know about that number it's smaller

719
00:45:00,329 --> 00:45:04,344
比lambda1或者它可能会击中lambda1
than lambda 1 or it might hit lambda 1

720
00:45:04,349 --> 00:45:06,445
但它并不大，这就是为什么千里马
but it's not bigger that's why Maxima

721
00:45:06,450 --> 00:45:09,415
很容易你放入任何矢量和你
are easy you put in any vector and you

722
00:45:09,420 --> 00:45:11,395
知道你知道发生了什么
know what's happening you know it

723
00:45:11,400 --> 00:45:13,584
不是它不高于最大值
doesn't it's it's not above the max

724
00:45:13,589 --> 00:45:19,064
很明显，什么是min什么的
obviously and what about the min what's

725
00:45:19,069 --> 00:45:22,254
这当然同样简单
that's equally simple of course it's at

726
00:45:22,259 --> 00:45:25,014
底部，所以这将是最小
the bottom so what would be the minimum

727
00:45:25,019 --> 00:45:29,004
如果我是那么真的那个商
of that really of that quotient if I was

728
00:45:29,009 --> 00:45:29,814
寻找
looking for

729
00:45:29,819 --> 00:45:32,155
我将得到什么特征向量和特征值
what eigen vector and eigen value will I

730
00:45:32,160 --> 00:45:36,024
发现当我看到这个，我的底部
find when I look at the bottom of this I

731
00:45:36,029 --> 00:45:41,395
会找到lambda和最后一个人lambda
will find lambda and the last guy lambda

732
00:45:41,400 --> 00:45:43,995
分
min

733
00:45:44,000 --> 00:45:47,265
我认为X将成为它的所在
and the I think where the X will be its

734
00:45:47,270 --> 00:45:50,325
特征向量再次这个东西会
eigenvector and again this stuff will

735
00:45:50,330 --> 00:45:56,025
相等的lambdan，所以我很容易知道
equal lambda n so that's easy I know

736
00:45:56,030 --> 00:46:00,455
如果我放任何矢量无论如何
that if I put in any vector whatever

737
00:46:00,460 --> 00:46:03,675
只需选择n维的任何向量
just choose any vector in n dimensions

738
00:46:03,680 --> 00:46:06,885
并计算X的X现在我也是
and compute R of X what do I now also

739
00:46:06,890 --> 00:46:09,785
知道我们的那个载体
know about our that our of that vector

740
00:46:09,790 --> 00:46:17,325
它比Lambda和下面的更大
it's greater than lambda and below the

741
00:46:17,330 --> 00:46:21,825
最大限度高于男性，但现在如果我投入
max above the men but now if I put in

742
00:46:21,830 --> 00:46:24,705
现在来谈谈其他lambda表达式以及
now what about the other lambdas well

743
00:46:24,710 --> 00:46:26,715
重点是那些是马鞍
the point is that those are saddle

744
00:46:26,720 --> 00:46:29,535
指出这个美丽的东西
points the beautiful thing about this

745
00:46:29,540 --> 00:46:32,085
瑞利商是它的衍生物
Rayleigh quotient is its derivative

746
00:46:32,090 --> 00:46:35,775
在T的鞍点处等于0
equals 0 right at the saddle point at T

747
00:46:35,780 --> 00:46:39,795
在特征向量和它的值
at the eigenvectors and and its value at

748
00:46:39,800 --> 00:46:43,155
特征向量就是特征值
the eigenvectors is the eigenvalue so so

749
00:46:43,160 --> 00:46:46,185
我中有你看到我在说什么，我有
i have you see what i'm saying i have

750
00:46:46,190 --> 00:46:50,685
lambda1这里是最大的lambda，这里是a
lambda 1 here a max lambda and here a

751
00:46:50,690 --> 00:46:53,325
我和他之间有一堆
min and in between i have a bunch of

752
00:46:53,330 --> 00:46:56,475
其他lambdas是马鞍点
other lambdas which are saddle points

753
00:46:56,480 --> 00:47:02,085
如果我把x放入x的2r和
and if i put an x in to the 2 r of x and

754
00:47:02,090 --> 00:47:05,085
看看会发生什么我不知道
look to see what happens i have no idea

755
00:47:05,090 --> 00:47:08,465
我是否在这一侧或下方
whether I'm on this side the below it or

756
00:47:08,470 --> 00:47:12,915
这边是lambda我是马鞍
this side above lambda I so the saddle

757
00:47:12,920 --> 00:47:16,875
积分更难，需要一个
points are more difficult and take a

758
00:47:16,880 --> 00:47:21,255
更多的耐心，所以这是另一个
little more patience so that's the other

759
00:47:21,260 --> 00:47:27,975
马鞍点的来源和我这些
source of saddle points and I so these

760
00:47:27,980 --> 00:47:32,505
是的，让我再强调一下
are so let me just emphasize again what

761
00:47:32,510 --> 00:47:39,215
我在λ等于QKI的lambda说
I'm saying at lambda at X equal Q K I

762
00:47:39,220 --> 00:47:47,495
有一些号码
have some number the

763
00:47:47,500 --> 00:47:52,645
我们的前任有一些积极的
our ex has some number of positive

764
00:47:52,650 --> 00:47:54,905
特征值和一些负数
eigenvalues and some number of negative

765
00:47:54,910 --> 00:48:00,545
适用于QK上下的事情
ones for the things above and below QK

766
00:48:00,550 --> 00:48:04,175
好吧我没时间跟进了
okay I've run out of time to to follow

767
00:48:04,180 --> 00:48:08,675
在那个鞍点上
up on the saddle point part of that on

768
00:48:08,680 --> 00:48:13,135
这张照片的细节我可能会
the details of this picture I might

769
00:48:13,140 --> 00:48:16,685
这将在笔记中，我可能会来
that'll be in the notes and I might come

770
00:48:16,690 --> 00:48:18,575
在下一个开始时回到它
back to it at the very start of next

771
00:48:18,580 --> 00:48:22,625
时间，但我会在此之前你会
time but I will before that you will

772
00:48:22,630 --> 00:48:30,785
有三个实验室然后我
have the lab number three and then I

773
00:48:30,790 --> 00:48:32,795
我认为我们应该讨论它，因为我
think we should discuss it because I

774
00:48:32,800 --> 00:48:37,865
没有完成这个实验室，这是它的意图
haven't done this lab it's it's intended

775
00:48:37,870 --> 00:48:40,535
给你一些过度拟合的感觉
to give you some feeling for overfitting

776
00:48:40,540 --> 00:48:42,875
并且还打算给你一点
and also intended to give you a little

777
00:48:42,880 --> 00:48:47,165
介绍深度学习等
introduction to deep learning and so

778
00:48:47,170 --> 00:48:50,315
我会告诉你的，我们可以谈谈
I'll get it to you and we can talk about

779
00:48:50,320 --> 00:48:53,525
星期三，它将不会到期
it Wednesday and again it won't be due

780
00:48:53,530 --> 00:48:56,580
until the Wednesday after break okay

781
00:48:53,530 --> 00:49:00,300
到休息后的星期三好吧

782
00:48:56,590 --> 00:49:00,300
谢谢，我星期三见
thanks so I'll see you Wednesday