3 - 6 - Inverse and Transpose (11 min).srt

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In this video, I want to
在这一讲中
(字幕整理：中国海洋大学 黄海广,haiguang2000@qq.com )

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tell you about a couple of special
我将介绍一些特殊的矩阵运算

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matrix operations, called the
也就是矩阵的逆运算

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matrix inverse and the matrix transpose operation.
以及矩阵的转置运算

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Let's start by talking about matrix
首先我们从逆矩阵开始

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inverse, and as
同往常一样

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usual we'll start by thinking about
我们依然先思考一下

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how it relates to real numbers.
矩阵运算和实数运算的关系

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In the last video, I said
在上一段视频中

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that the number one plays the
我讲过 在实数空间中

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role of the identity in
数字1扮演了单位矩阵的角色

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the space of real numbers because
因为1和任何数相乘

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one times anything is equal to itself.
其结果都是那个数本身

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It turns out that real numbers
我们都知道

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have this property that very
实数有这样的一个性质

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number have an, that
那就是每一个实数

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each number has an inverse,
都有一个倒数

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for example, given the number
举个例子

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three, there exists some
对于数字3

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number, which happens to
一定存在某个数

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be three inverse so that
是3的倒数

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that number times gives you
这个倒数乘以3

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back the identity element one.
其乘积将得到单位元1

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And so to me, inverse of course this is just one third.
当然 这里的逆也就是三分之一

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And given some other number,
而对于另一个数

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maybe twelve there is
比如说12

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some number which is the
那么一定有某个数

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inverse of twelve written as
是12的倒数

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twelve to the minus one, or
也可以写作12的-1次方

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really this is just one twelve.
其实也就是1/12

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So that when you multiply these two things together.
因此当你将这两个数相乘的时候

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the product is equal to
其结果依然是等于1

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the identity element one again.
或者可以称为单位元1

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Now it turns out that in
然而事实上 在实数空间内

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the space of real numbers, not everything has an inverse.
并非所有实数都有倒数

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For example the number zero
比如说数字0就没有倒数

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does not have an inverse, right?
是吧？

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Because zero's a zero inverse, one over zero that's undefined.
因为0的倒数 也就是1/0 是没有意义的

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Like this one over zero is not well defined.
像这里 1除以0是未被定义的

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And what we want to
在这一节剩下的内容中

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do, in the rest of this
我们将要解决的一个问题

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slide, is figure out what does
是求解一个矩阵的逆

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it mean to compute the inverse of a matrix.
是什么意思

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Here's the idea: If
概念如下

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A is a n by
如果A是一个矩阵

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n matrix, and it
其维度为m行m列

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has an inverse, I will say
那么A矩阵有其逆矩阵

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a bit more about that later, then
后面我还会详细介绍这一点

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the inverse is going to
那么这个逆矩阵

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be written A to the
可以写成

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minus one and A
矩阵A的-1次方

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times this inverse, A to
同时 矩阵A乘以它的逆矩阵

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the minus one, is going to
A的-1次方

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equal to A inverse times
也等于A的-1次方乘以A

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A, is going to
其结果将等于

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give us back the identity matrix.
单位矩阵

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Okay?
对吧？

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Only matrices that are
只有维度是m行m列的矩阵

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m by m for some the idea of M having inverse.
才有其逆矩阵

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So, a matrix is
因此

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M by M, this is also
如果一个矩阵的维度是m行m列

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called a square matrix and
那么这个矩阵也可以称之为方阵

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it's called square because
称其为方阵是因为

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the number of rows is equal to the number of columns.
这类矩阵的行数和列数相等

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Right and it turns out
好的

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only square matrices have inverses,
实际上只有方阵才有逆矩阵

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so A is a square
所以 如果A是一个方阵

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matrix, is m by m,
其维度是m行m列

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on inverse this equation over here.
那么它将满足这个等式

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Let's look at a concrete example,
接下来我们看一个具体的例子

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so let's say I
假设说

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have a matrix, three, four,
我们有这样一个矩阵 3 4

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two, sixteen.
2 16

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So this is a two by
这是一个2行2列的矩阵

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two matrix, so it's
因此这个矩阵

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a square matrix and so this
是一个方阵

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may just could have an and
因此这个矩阵可以有它的逆矩阵

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it turns out that I
假如说

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happen to know the inverse
这个矩阵的逆矩阵是

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of this matrix is zero point
0.4

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four, minus zero point
-0.1

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one, minus zero point
-0.05

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zero five, zero zero seven five.
0.075

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And if I take this matrix
那么如果我用这个逆矩阵

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and multiply these together it
和原来的矩阵相乘

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turns out what I get
那么

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is the two by
我们将得到的结果

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two identity matrix, I,
是一个2行2列的单位矩阵I

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this is I two by two.
这就是矩阵I 维度是2行2列

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Okay?
没问题吧？

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And so on this slide,
在这张幻灯片上

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you know this matrix is
这个矩阵就是矩阵A

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the matrix A, and this matrix is the matrix A-inverse.
这个矩阵就是A的逆矩阵

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And it turns out
结果是

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if that you are computing A
如果你要计算A乘以A的逆矩阵

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times A-inverse, it turns out
或者说

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if you compute A-inverse times
A的逆矩阵乘以A

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A you also get back the identity matrix.
你将得到一个单位矩阵

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So how did I
那么

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find this inverse or how
怎样得到这个逆矩阵呢

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did I come up with this inverse over here?
或者说我是怎么知道这个逆矩阵的呢？

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It turns out that sometimes
实际上有时候你可以

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you can compute inverses by hand
自己用笔算出来

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but almost no one does that these days.
但可能现在没多少人这么求逆矩阵了

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And it turns out there is
实际上我们有很多很好的软件

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very good numerical software for
可以用来进行数学运算

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taking a matrix and computing its inverse.
能很容易地对矩阵进行求逆运算

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So again, this is one of
因此 同样地

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those things where there are lots
对于这个问题

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of open source libraries that
你可以在很多主流的编程环境中实现

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you can link to from any
这些环境一般都有很多开源库

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of the popular programming languages to compute inverses of matrices.
你可以直接运用来求解逆矩阵

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Let me show you a quick example.
这里我举一个简单的例子

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How I actually computed this inverse,
来说明一下怎样求逆矩阵

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and what I did was I used software called Optive.
我将使用一个叫Octave的软件

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So let me bring that up.
打开这个软件

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We will see a lot about Optive later.
之后我们还会更多地用到Octave

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Let me just quickly show you an example.
这里我只是很快地举一个例子

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Set my matrix A to
定义一个矩阵A

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be equal to that matrix on
对它赋值为左边幻灯片上的矩阵

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the left, type three four
键入3 4 2 16

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two sixteen, so that's my matrix A right.
这就是我的A矩阵了

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This is matrix 34,
这就是矩阵 3 4

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216 that I have down
2 16

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here on the left.
也就是左边那个矩阵

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And, the software lets me compute
使用这个软件

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the inverse of A very easily.
我能够很容易得到A的逆矩阵

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It's like P over A equals this.
就像这样直接键入pinv(A)

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And so, this is right,
这样

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this matrix here on my
我们得到了结果

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four minus, on my one, and so on.
0.4 -0.1 等等

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This given the numerical
这里得到的是

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solution to what is the
A的逆矩阵的近似解

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inverse of A. So let me
我可以这样写

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just write, inverse of A
定义一个变量inverseOFA

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equals P inverse of
其值等于 pinv(A)

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A over that I
那么inverseOFA的值就是这样

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can now just verify that A
现在我们可以证明一下

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times A inverse the identity
A乘以A的逆矩阵结果是单位矩阵

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is, type A times the
键入 A 乘以

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inverse of A and
A的逆矩阵

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the result of that is
这样得到的结果

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this matrix and this is
是这样的一个矩阵

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one one on the diagonal
主对角线是1和1

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and essentially ten to
副对角线不是1

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the minus seventeen, ten to the
但是基本上10的-17次方

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minus sixteen, so Up to
10的-16次方

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numerical precision, up to
这里由于计算精度的问题

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a little bit of round off
由于计算机在寻找最佳结果时

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error that my computer
由于进行了四舍五入圆整

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had in finding optimal matrices
所以产生了一点点误差

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and these numbers off the
所以这些副对角线上的数字

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diagonals are essentially zero
实际上也近似等于0

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so A times the inverse is essentially the identity matrix.
因此矩阵A和其逆矩阵相乘的结果就是单位矩阵

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Can also verify the inverse of
我们也可以证明

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A times A is also
A的逆矩阵乘以A

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equal to the identity,
其结果也是单位矩阵

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ones on the diagonals and values
主对角线上都是1

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that are essentially zero except
副对角线上的数有小数

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for a little bit of round
但四舍五入圆整一下

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dot error on the off diagonals.
实际上其值也等于0

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If a definition that the inverse
关于逆矩阵的定义

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of a matrix is, I had
我需要强调一点

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this caveat first it must
首先

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always be a square matrix, it
矩阵必须是方阵

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had this caveat, that if
注意这里

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A has an inverse, exactly what
如果A有其逆矩阵

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matrices have an inverse
究竟什么矩阵有其逆矩阵的问题

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is beyond the scope of this
已经超出了这节线性代数复习课

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linear algebra for review that one
所讨论的范畴

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intuition you might take away
但有一点

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that just as the
你能理解的是

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number zero doesn't have an
数字0没有倒数

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inverse, it turns out
因此

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that if A is say the
如果矩阵A中

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matrix of all zeros, then
所有元素都为0

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this matrix A also does
那么这个矩阵A

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not have an inverse because there's
依然没有逆矩阵

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no matrix there's no A
因为没有这样的A矩阵

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inverse matrix so that this
能使得这个矩阵

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matrix times some other
乘以另一个矩阵

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matrix will give you the
其值能得到单位矩阵

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identity matrix so this matrix of
所以

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all zeros, and there
所有元素都为0的矩阵

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are a few other matrices with properties similar to this.
以及一些其他类似这样的矩阵

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That also don't have an inverse.
它们都没有逆矩阵

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But it turns out that
但是实际上

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in this review I don't
在这节复习课中

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want to go too deeply into what
我不想太深入地介绍

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it means matrix have an
矩阵的逆矩阵有何意义等问题

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inverse but it turns
但是实际上

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out for our machine learning
对于我们机器学习的应用来讲

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application this shouldn't be
这一点不应成为问题

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an issue or more precisely
具体来讲

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for the learning algorithms where
对于某种机器学习算法

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this may be an to namely
可能会碰到这种问题的讨论

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whether or not an inverse matrix
是否存在逆矩阵这样的问题

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appears and I will tell when
在我们碰到这种学习算法时

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we get to those learning algorithms
我再告诉你

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just what it means for an
一种算法到底有没有逆矩阵

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algorithm to have or not
究竟应该怎样理解

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have an inverse and how to fix it in case.
以及应该怎样解决具体的问题

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Working with matrices that don't
比如怎样处理

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have inverses.
矩阵的逆矩阵不存在的问题

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But the intuition if you
不过你至少可以这样理解

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want is that you can
在某种意义上

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think of matrices as not
你可以把那些

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have an inverse that is somehow
没有逆矩阵的矩阵

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too close to zero in some sense.
想成是非常近似为0

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So, just to wrap
最后再讲一点

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up the terminology, matrix that
逆矩阵不存在的矩阵

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don't have an inverse Sometimes called
它的专有名词是

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a singular matrix or degenerate
奇异矩阵

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matrix and so this
或者叫退化矩阵

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matrix over here is an
因此这里这个矩阵

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example zero zero zero matrix.
这个零矩阵

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is an example of a matrix that is singular, or a matrix that is degenerate.
就是一个奇异矩阵 或者说退化矩阵的例子

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Finally, the last special
最后 还有一种矩阵运算

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matrix operation I want to
我想给大家介绍的

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00:07:32,652 --> 00:07:34,520
tell you about is to do matrix transpose.
是矩阵的转置运算

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So suppose I have
假设我们有矩阵A

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matrix A, if I compute
那么A的转置矩阵

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the transpose of A, that's what I get here on the right.
可以写成右边这个矩阵

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This is a transpose which is
这就是矩阵A的转置矩阵

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written and A superscript T,
写法是A加上一个上标的T

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and the way you compute
计算转置矩阵的方法

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the transpose of a matrix is as follows.
如下所示

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To get a transpose I am going
要得到转置矩阵

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to first take the first
首先我取出A矩阵的第一行

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row of A one to zero.
1 2 0

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That becomes this first column of this transpose.
它们将成为转置矩阵的第一列

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And then I'm going to take
接下来

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the second row of A,
我再取出矩阵A的第二行

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3 5 9, and that becomes the second column.
3 5 9

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of the matrix A transpose.
它们将成为转置矩阵的第二列

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And another way of
你也可以这样想

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thinking about how the computer transposes
求解转置矩阵

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is as if you're taking this
实际上可以看作

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sort of 45 degree axis
画一条45度的斜线

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and you are mirroring or you
然后你以这条线求镜像

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are flipping the matrix along that 45 degree axis.
或者以这条45度线为轴进行翻转

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so here's the more formal
因此我们可以得到

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definition of a matrix transpose.
关于矩阵转置的正式定义

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Let's say A is a m by n matrix.
假设矩阵A是一个m行n列的矩阵

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And let's let B equal A
并且假设矩阵B等于A的转置

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transpose and so BA transpose like so.
就像这样

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Then B is going to
那么B将是一个

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be a n by m matrix
n行m列的矩阵

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with the dimensions reversed so
两个矩阵的维度是相反的

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here we have a 2x3 matrix.
这里我们的A矩阵是2行3列的

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And so the transpose becomes a
那么A的转置矩阵B

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3x2 matrix, and moreover,
将是一个3行2列的矩阵

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the BIJ is equal to AJI.
此外 Bij等于Aji

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So the IJ element of this
也就是说

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matrix B is going to be
矩阵B的ij元素

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the JI element of that
是等于A矩阵的ji元素

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earlier matrix A. So for
我们来举个例子

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example, B 1 2
B12这个元素

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00:09:04,212 --> 00:09:06,997
is going to be equal
将等于

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to, look at this
请看这个矩阵

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matrix, B 1 2 is going to be equal to
B12元素将等于

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this element 3 1st row, 2nd column.
A矩阵中 第一行第二列的元素3

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And that equal to this, which
那么这个元素

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is a two one, second
也就是等于A21

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row first column, right, which
A矩阵中第二行第一列的元素

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is equal to two and some
也就是2

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of the example B 3
再举个例子

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2, right, that's B
B32这个元素

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3 2 is this element 9,
也就是9

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and that's equal to
是等于

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a two three which is
A23这个元素

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this element up here, nine.
也就是这个元素 也是等于9的

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And so that wraps up
好的

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the definition of what it
这样我们就介绍了

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means to take the transpose
怎样求一个矩阵的转置矩阵

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of a matrix and that
以及它的正式定义

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in fact concludes our linear algebra review.
这也是我们的线性代数复习课的最后一节

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So by now hopefully you know
现在你应该已经掌握

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how to add and subtract
如何进行矩阵的加法运算

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00:09:52,205 --> 00:09:53,701
matrices as well as
以及乘法运算

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00:09:53,701 --> 00:09:55,325
multiply them and you
同时

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also know how, what are
你也应该了解

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00:09:57,185 --> 00:09:58,942
the definitions of the inverses
矩阵的逆运算

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00:09:58,942 --> 00:10:01,457
and transposes of a matrix
和转置运算的定义

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00:10:01,457 --> 00:10:02,934
and these are the main operations
这些都是这门课中

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used in linear algebra
需要用到的线性代数中

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00:10:05,152 --> 00:10:06,172
for this course.
最基本的一些运算

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In case this is the first time you are seeing this material.
也许这是你第一次观看我们的视频

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00:10:09,043 --> 00:10:10,798
I know this was a lot
我知道这里一下子讲了

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00:10:10,798 --> 00:10:13,032
of linear algebra material all presented
很多线性代数的知识

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00:10:13,032 --> 00:10:14,512
very quickly and it's a
讲的很多也讲的很快

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00:10:14,520 --> 00:10:16,581
lot to absorb but
可能需要一点时间来消化

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00:10:16,581 --> 00:10:18,102
if you there's no need
你没有必要记住

294
00:10:18,102 --> 00:10:20,045
to memorize all the definitions
我们刚刚讲过的

295
00:10:20,045 --> 00:10:21,718
we just went through and if
所有那些定义

296
00:10:21,718 --> 00:10:23,451
you download the copy of either
如果你从课程网站上

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00:10:23,451 --> 00:10:24,520
these slides or of the
下载了这些幻灯片

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00:10:24,540 --> 00:10:28,353
lecture notes from the course website.
或者讲义

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00:10:28,370 --> 00:10:29,645
and use either the slides or
并且使用它们

300
00:10:29,645 --> 00:10:31,478
the lecture notes as a reference
作为一个参考内容

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00:10:31,490 --> 00:10:32,886
then you can always refer back
那么你可以随时返回

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00:10:32,900 --> 00:10:34,178
to the definitions and to figure
查看这些定义和概念

303
00:10:34,178 --> 00:10:35,615
out what are these matrix
随时理解这些矩阵的乘法 转置

304
00:10:35,615 --> 00:10:39,111
multiplications, transposes and so on definitions.
等等这些运算的概念

305
00:10:39,140 --> 00:10:40,697
And the lecture notes on the course website also
同时 课程官网的讲义中

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00:10:40,697 --> 00:10:42,421
has pointers to additional
我们也提供了链接

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00:10:42,450 --> 00:10:44,675
resources linear algebra which
也是一些其他的线性代数资料

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00:10:44,675 --> 00:10:47,445
you can use to learn more about linear algebra by yourself.
你可以自己点击学习

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00:10:48,861 --> 00:10:53,445
And next with these new tools.
接下来 运用这些所学的工具

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00:10:53,540 --> 00:10:55,153
We'll be able in the next
在接下来的几段视频中

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00:10:55,153 --> 00:10:57,035
few videos to develop more powerful
我们将介绍

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00:10:57,035 --> 00:10:58,758
forms of linear regression that
非常重要的线性回归

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00:10:58,758 --> 00:10:59,854
can view of a lot
我们会看到更多的数据

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00:10:59,854 --> 00:11:00,809
more data, a lot more
更多的特征

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00:11:00,809 --> 00:11:02,226
features, a lot more training
以及更多的训练样本

316
00:11:02,226 --> 00:11:04,367
examples and later on
再往后

317
00:11:04,400 --> 00:11:06,114
after the new regression we'll actually
在介绍线性回归之后

318
00:11:06,114 --> 00:11:07,832
continue using these linear
我们还将继续使用这些线代工具

319
00:11:07,832 --> 00:11:10,016
algebra tools to derive more
来推导一些

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00:11:10,016 --> 00:11:13,242
powerful learning algorithims as well
更加强大的学习算法