10 Example Transposes在线视频

好了
All right. So let's talk about this

今天我们讨论一种特殊类型的矩阵
particular type of matrix.

对称矩阵
You can say

对称矩阵
really symmetric matrix.

在讲对称矩阵之前
But before we discuss symmetric matrix，

我们先介绍一个概念
we need to introduce a concept

矩阵的转置
so called transpose of a matrix.

看这里
Here we go.

这个m乘n矩阵A的转置定义如下
Here the transpose of an

这个m乘n矩阵A的转置定义如下
m by a matrix A is an n by n

这个m乘n矩阵A的转置定义如下
matrix b defined by the following.

a_ij=b_ji 注意到下标互换了
So you can have a_ij and

a_ij=b_ji 注意到下标互换了
b j I so you can see are

a_ij=b_ji 注意到下标互换了
you're interchange this two indices.

得到了另外一个矩阵
And you are having another matrix.

得到了另外一个矩阵
For i, or you can say from

I从1到m
one to m, j from one to n，

j从1到n
here we need to make sure that

i代表矩阵A的元素的行下标
you are having i, which is is

i代表矩阵A的元素的行下标
really the row number.

I从1变到m
So which is 1,2,, m,

j从1到n 行列互换后
this is 1,2,n.

j从1到n 行列互换后
Are we clear?

这里得到的是一个新的矩阵
So now you're having like two indices

这里得到的是一个新的矩阵
switched，

是原来矩阵的转置
so you'd be able to get a matrix,

是原来矩阵的转置
which is so called the transpose

是原来矩阵的转置
of the original matrix.

A的转置记为A^T
The transpose of A is denoted

看一下这个具体的例子
by A transpose. Somehow you can just

看一下这个具体的例子
look at this the particular example,

这是一个3乘2的矩阵
you are having some three

这是一个3乘2的矩阵
by two matrix here:

1 2 3 4 5 6
1,2;3,4;5,6.

也就是有3行和两列 转置过后
So you are having three rows

也就是有3行和两列 转置过后
and two columns.

也就是有3行和两列 转置过后
When you take the transpose，

就得到这个矩阵 1 2 3 4 5 6
you will be able to have this

就得到这个矩阵 1 2 3 4 5 6
kind of matrix 1,2,3;

就得到这个矩阵 1 2 3 4 5 6
and 4,5,6.

也就是说第一列变成了第一行
Are we clear?

也就是说第一列变成了第一行
So you can see the first column

第二列变成第二行
becomes the first row，

第二列变成第二行
and the second column

第二列变成第二行
becomes the second row.

得到了矩阵A的转置
So you are just having this

这个第一行呢
kind of A transpose.

变成第二个矩阵的第一列
You can say，what about the first row？

变成第二个矩阵的第一列
It becomes the first column

变成第二个矩阵的第一列
of the second matrix.

可以在看一些其它矩阵的情况
Somehow you can just understand

可以在看一些其它矩阵的情况
better by working out other matrices，

具体理解一下矩阵的转置
to understand better

具体理解一下矩阵的转置
about the transpose of a matrix.

可以考虑长方形矩阵的转置
You can say, here we allow triangular,

可以考虑长方形矩阵的转置
somehow you can say rectangular

也可以考虑方阵的转置
matrix definitely.

也可以考虑方阵的转置
What about the square matrix?

也可以考虑方阵的转置
You can do so.

看这个2乘2的矩阵
Like if you take B:

a b c d 它是一个方阵
a,b;c,d, for example，

a b c d 它是一个方阵
like two by two matrix，

a b c d 它是一个方阵
which is a square matrix.

它的逆矩阵是什么样子呢
Now，what about B transpose?

是下面这个矩阵
It becomes the following one.

是下面这个矩阵
You can say， a，b，

这时候a b原来是第一行 变成第一列
which is the first row，

这时候a b原来是第一行 变成第一列
becomes the first column.

第二行呢
What about the second row？

变成了第二列
Second row becomes the

变成了第二列
second column.

所以我们有了这个
And definitely you are having this.

也就说
That means the transpose,

在考虑矩阵的转置的时候
you can say,for the definition.

允许矩阵是长方形或者是方阵
We consider somehow matrix，

允许矩阵是长方形或者是方阵
which could be rectangular

允许矩阵是长方形或者是方阵
matrix or square matrix.

现在我们来看一下转置
No problem.

现在我们来看一下转置
Now we want to understand

现在我们来看一下转置
better about the connections between

和其它知识点之间的联系
other topics we introduced earlier like,

和其它知识点之间的联系
what about somehow you can say,

首先是对称矩阵.
conclusion about symmetric matrix.

如何考虑对称矩阵呢
You probably may say,

如何考虑对称矩阵呢
what about the matrix?

这个是很重要的
Actually，

这个是很重要的
which is somehow very

这个是很重要的
important for us to discuss.

看一下这个矩阵A和它的转置
So if you look at this A

看一下这个矩阵A和它的转置
and A tanspose，

如果它们是一样的
if they are the same

如果它们是一样的
and we call this matrix symmetric.

那么它就是一个对称矩阵
So that means A is a symmetric.

这就是对称矩阵的定义
Somehow you can have this

这就是对称矩阵的定义
kind of definition: symmetric matrix.

A等于A的转置
So you take the transpose of A

A等于A的转置
and it's the same as the matrix A

这个矩阵A就是对称矩阵
and this is really some kind

这个矩阵A就是对称矩阵
of symmetric matrix.

这里有一个例子
Let me give you some kind of example.

-1 2 0 2 -1
Over here

-1 2 0 2 -1
you can have a negative one，

-1 2 0 2 -1
two，and zero to negative one.

0 0 0 1
Somehow you can say, 0,0,1.

这就是一个对称矩阵
So this is a really symmetric matrix，

这就是一个对称矩阵
because when you take the

因为这个矩阵和它的转置矩阵相等
so called transpose of this matrix，

因为这个矩阵和它的转置矩阵相等
and nothing change it.

这就说明这个A矩阵
That means A transpose

这就说明这个A矩阵
is actually the same as A,

这也是我们需要
which is really the so called

这也是我们需要
symmetric matrix. Actually，

讨论转置的主要原因之一
which is one of the main purposes

讨论转置的主要原因之一
for us to discuss transpose.

转置的概念直接
Ok. Transposes allow us t

道出了对称矩阵的概念
o define symmetric matrices.

这些对于我们
And those are important

考虑实际问题都很有帮助
for us to model practical problems

考虑实际问题都很有帮助
in our future discussions.

看一下这个m乘n的
As you can see actually，here，

看一下这个m乘n的
whenever you have some

长方形矩阵 R
kind of rectangular matrix，

长方形矩阵 R
maybe we could denote it by R，

长方形矩阵 R
which is somehow you

长方形矩阵 R
can say m by n matrix.

这里m不一定等于n
Here，m is not necessarily

这里m不一定等于n
equal to n. m is not necessarily

也就是说这个矩阵可能是长方形的
equal to n, so that's why this matrix

也就是说这个矩阵可能是长方形的
may be rectangular.

然而 如果我们考虑RR^T和
However，when you take this R

然而 如果我们考虑RR^T和
transpose R,

R^TR 就得到了两个对称矩阵
or R R transpose,

R^TR 就得到了两个对称矩阵
they are both symmetric matrices.

同学们可以自己验证一下
You can double check,

同学们可以自己验证一下
because when you take the

同学们可以自己验证一下
of each of the matrix,

同学们可以自己验证一下
and you will be getting

同学们可以自己验证一下
back to itself， right？

这里强调一下
But here，

这里强调一下
let me just mention something here.

这里强调一下
So if you take look at this A

对AB求转置得到的是 B^TA^T
multiplied by B, and take the transpose，

对AB求转置得到的是 B^TA^T
you will be getting somehow

对AB求转置得到的是 B^TA^T
the following one: B transpose

注意一下 顺序变了
A transpose.

注意一下 顺序变了
This kind of order is reversed.

这个结论和求乘积矩阵的逆矩阵类似
Very much like this,

这个结论和求乘积矩阵的逆矩阵类似
and you would be able to have

这个结论和求乘积矩阵的逆矩阵类似
some kind of similar conclusion

这个结论和求乘积矩阵的逆矩阵类似
regarding the inverse of a matrix.

如果A和B都是可逆的矩阵
Suppose A and B are invertible.

如果A和B都是可逆的矩阵
All right.

如果A和B都是可逆的矩阵
So suppose you are having two

如果A和B都是可逆的矩阵
invertible matrices.

现在考虑它们的乘积的逆矩阵
Now if you take the product of these two,

现在考虑它们的乘积的逆矩阵
and definitely you can

现在考虑它们的乘积的逆矩阵
take the inverse of this product.

大家可以得到这个
You would be able to get

大家可以得到这个
somehow like this:

B^-1A^-1
B inverse and A inverse,

B^-1A^-1
very much like this.

同学们可以考虑一下具体的细节
You guys can check out all the details

同学们可以考虑一下具体的细节
and understand why we are

同学们可以考虑一下具体的细节
having these two conclusions.

用这个结论
And definitely，

就能够很好地说明这两个矩阵是对称的
you can apply this kind of conclusion

就能够很好地说明这两个矩阵是对称的
to these two scenarios,

就能够很好地说明这两个矩阵是对称的
to understand why we are having this two

就能够很好地说明这两个矩阵是对称的
symmetric matrices.

就能够很好地说明这两个矩阵是对称的
Very much like this.

在第三章里面
We will be talking about these

在第三章里面
two matrices

在第三章里面
extensively in chapter number three.

我们会详细讨论这两个矩阵
So we will be doing that later.

我们会详细讨论这两个矩阵
Ok，here we go.

如果有了这两个定义
If you are having somehow the

如果有了这两个定义
discussions about these two definitions.

如果有了这两个定义
First definition is like

转置矩阵和对称矩阵
so called transpose of a matrix.

转置矩阵和对称矩阵
The second one is really

转置矩阵和对称矩阵
so called symmetric matrix.

我们可以来考虑一个具体的矩阵A
Now we can take a look at some

我们可以来考虑一个具体的矩阵A
kind of particular example，

我们可以来考虑一个具体的矩阵A
A, which is given by the following

它的第一行是2 -1 0 0
one like: two,

它的第一行是2 -1 0 0
negative 1,0,0.

这个矩阵是一个很有趣的矩阵
This is really some very interesting

这个矩阵是一个很有趣的矩阵
matrix we can talk about in great detail.

值得我们好好研究
And negative one，two,

第二行是-1 2 -1 0
-1,0. All right. So this is really negative 1,one more time.

后面两行是0
This is zero，negative 1,2，

-1 2 -1 0 0 -1 2
negative one，

-1 2 -1 0 0 -1 2
0,0,-1,2.

这个矩阵是对称的吗
So can I ask you guys whether this

这个矩阵是对称的吗
matrix is symmetric or not.

同学们可以根据定义验证一下
Maybe you can say I can check this:

A=A^T 这说明A是对称的矩阵
A transpose is equal to A,definitely,

A=A^T 这说明A是对称的矩阵
check. So that means A is symmetric.

再次说明一下
A is symmetric according to the definition

再次说明一下
we introduced earlier. Actually，

A等于它的转置
you are having the transpose

A等于它的转置
equals the original matrix.

我们知道了这个矩阵是对称的
Therefore，you can say this matrix

我们知道了这个矩阵是对称的
is symmetric. Somehow you can say,

接下来我们考虑它的
what about if you consider like somehow

接下来我们考虑它的
LU factorization or LDU factorization？

LU分解和LDU分解
You can work out somehow the

LU分解和LDU分解
LDU factorization for this matrix.

这个矩阵是4乘4的
But really here this

这个矩阵是4乘4的
is a 4 by 4 matrix.

留给大家做作业
We can take a look at like somehow

现在考虑一个类似的3乘3的矩阵
you can say three by three matrix，

现在考虑一个类似的3乘3的矩阵
maybe I can just say

现在考虑一个类似的3乘3的矩阵
what about this matrix？

考虑这个简单一些的情况
We consider somehow simple

考虑这个简单一些的情况
situation like three by three matrix:

2 -1 0 -1 2 -1 0- 1 2
two，negative 1, 0，negative 1,2，

2 -1 0 -1 2 -1 0- 1 2
negative one，and zero，

2 -1 0 -1 2 -1 0- 1 2
negative 1,2.

现在你有了这个矩阵
Now you’re having this matrix.

现在你有了这个矩阵
And how would you be able to figure out

你怎么才能算出我们
somehow the LDU factorization

之前提到的 ldu 因子分解
as we mentioned earlier.

可以怎么做呢
And definitely, you can do what?

可以做这个高斯消元的步骤
You can do the following Gaussian

从而把这个矩阵化为上三角矩阵
elimination steps and to convert this

从而把这个矩阵化为上三角矩阵
matrix into an upper triangular matrix.

好了
And eventually，let me just give you this

看这个具体的结果
particular result.

第一行是 1 0 0
And you will be able to have 1,0,0.

第二行是1/2 1 0
And negative one half,

第三行是
1 0. So that's for sure.

第三行是
And you’re having zero，

0 3/2 1 这个矩阵是L
negative 3/2，one. A

0 3/2 1 这个矩阵是L
ctually，this is L.

中间这个D矩阵是一个对角矩阵
And what about somehow

中间这个D矩阵是一个对角矩阵
you can say D in the middle，

中间这个D矩阵是一个对角矩阵
so which is really a diagonal matrix.

对角元为2 3/2 4/3
You are having two，

对角元为2 3/2 4/3
3/2, and 4/3.

非对角元都为零
You're having all off

非对角元都为零
diagonal entries zeros.

U是什么呢
And an interesting thing

U是什么呢
is like this.

U是什么呢
How would you be able to determine U？

U的第一行是1 -1/2 0
But look at this

U的第一行是1 -1/2 0
you would be able to have one,

U的第一行是1 -1/2 0
negative one half, zero.

第二行是0 1 -3/2
And what about this 0,1

第三行是
over here，which is a really negative 3/2,

0 0 1
and 0,0,1.

这个是中间的D矩阵
Ok. Now you can just try to

这个是中间的D矩阵
see here，which is in the middle D，

这个是U矩阵 U等于L的转置
this is U. But U is actually equal

看明白了吧
to L^T. Are we clear？

未来 如果有了一个对称矩阵
In the future，whenever you have a

未来 如果有了一个对称矩阵
symmetric matrix，

我们只要得到了U矩阵
when you consider this kind

我们只要得到了U矩阵
of LDU factorization，

转置一下就得到L矩阵
actually， you don't need to

转置一下就得到L矩阵
work out U, or you can say,

转置一下就得到L矩阵
you don't need to work out

转置一下就得到L矩阵
this L，specifically.

为什么呢
Why？

因为L等于U的转置
Because L is actually equal to

因为L等于U的转置
U transpose.

U等于L的转置
So that means you can say

U等于L的转置
U equals L transpose.

U等于L的转置
And you can say U

U等于L的转置
will be L transpose, here.

大家应该已经发现了
This is pretty much you can say,

如果考虑的矩阵是对角矩阵
if you are having symmetric matrix，

那么计算就会简单一些
many things may be simpler.

这个结论可以记下来
And of course，

这个结论可以记下来
it would be very good the thing

这个结论可以记下来
we need to keep in mind to understand

以后可以帮助我们理解矩阵的性质
better about matrices in the future.

以后可以帮助我们理解矩阵的性质
Ok.

大家可以问一个问题
Now you can just ask some

如果同时考虑逆矩阵和转置矩阵
questions like this：

如果同时考虑逆矩阵和转置矩阵
How would you able to consider somehow

会得到什么结论
you can say inverse

会得到什么结论
and the transpose together？

事实上
Actually，

这里有一个很有趣的结论
we are having a very interesting

A的逆矩阵的转置
result like A inverse transpose

等于A的转置的逆矩阵
is the same as a transpose

等于A的转置的逆矩阵
and inverse.

等于A的转置的逆矩阵
Are you clear?

也就说如果有一个对称的可逆矩阵
So that means you can say

也就说如果有一个对称的可逆矩阵
whenever you have a somehow

也就说如果有一个对称的可逆矩阵
invertible matrix，

它的逆矩阵也是对称的
which is also symmetric，

它的逆矩阵也是对称的
you can say what you can say

它的逆矩阵也是对称的
the inverse is also symmetric.

这个结论直接导出了这个结论
You can say，

这个结论直接导出了这个结论
this immediately follows from this

也就是说如果
kind of conclusion.

一个矩阵是对称可逆的
It means that whenever A is

它的逆矩阵也是如此
invertible and symmetric，

它的逆矩阵也是如此
so does its inverse.

它的逆矩阵也是如此
Very much like this.

仔细看一下这里
Look at this very carefully.

仔细看一下这里
You can say，

我们得到了这个很有趣的结论
You can say，

我们得到了这个很有趣的结论
how would you able to verify this

我们得到了这个很有趣的结论
equality or you can say this conclusion？

大家可以考虑一下原因
I will leave this as an exercise

大家可以考虑一下原因
to you guys.

大家可以考虑一下原因
All right.

这里有一个很好的练习题
Here is a very good spot check:

找到如下矩阵的
find the inverse and the

逆和它的LDU分解
LDU factorization

逆和它的LDU分解
of the following matrix.

这个矩阵是 2 1 1 1 2 1
Like 2,1,1;1,2,1;

1 1 2 它是一个对称矩阵
and 1,1,2. Of course

1 1 2 它是一个对称矩阵
here, you are looking at

1 1 2 它是一个对称矩阵
some symmetric matrix.

给大家几分钟的时间考虑这个问题
I will give you guys several minutes

过后大家可以一块讨论一下答案.
to work out this problem.

过后大家可以一块讨论一下答案.
And the solution will be

过后大家可以一块讨论一下答案.
provided shortly.

谢谢大家
Thank you.

大家应该都做了一下
I think everyone has the solution

大家应该都做了一下
to the problem we mentioned earlier。

刚刚的那道题目
When you go through this

做过之后应该对我们所讲的
kind of procedure，you can understand

做过之后应该对我们所讲的
better about all this kind of discussions

只是有了更深的理解
we talked about earlier？

我们这节课
Here we go.

首先介绍了矩阵的转置的概念
In this lecture，

首先介绍了矩阵的转置的概念
we introduce that it's

首先介绍了矩阵的转置的概念
kind of a definition，

首先介绍了矩阵的转置的概念
the transpose of a matrix.

同时也说明了我们介绍这个
And you can even think about

同时也说明了我们介绍这个
why we need to talk about this

概念的主要目的之一
definition. One of the main purposes

就是给出了对称矩阵的定义
is to consider symmetric matrix.

我们知道对称矩阵很重要
We know that symmetric matrix

我们知道对称矩阵很重要
is kind of critical for our future

我们知道对称矩阵很重要
discussions, and it will help us to

它将对数学建模起到关键的作用
model many important phenomena

它将对数学建模起到关键的作用
in our real life. Ok.

这类矩阵在后续的章节
Actually here,

这类矩阵在后续的章节
you can say this kind of matrices would be

中会继续性地用到
discussed extensively in our f

特别是第三章
uture chapters，especially

特别是第三章
in chapter number three.

如果有了这个定义
When you have this definition.

可以接着考虑它的性质
And you can talk about this kind of

可以接着考虑它的性质
properties like when you take the

比如这两个矩阵的乘积的转置矩阵
product of these two matrices

比如这两个矩阵的乘积的转置矩阵
and how would you be able to

比如这两个矩阵的乘积的转置矩阵
take the transpose？

应该等于B的转置乘以A的转置
And the transpose has to be,

应该等于B的转置乘以A的转置
somehow you can say，

应该等于B的转置乘以A的转置
given by B transpose times A transpose，

如果A等于A的转置
basically.

如果A等于A的转置
If A is equal to A transpose，

那么这个矩阵就是对称的
and this matrix is so called

那么这个矩阵就是对称的
symmetric matrix.

这是一个很好的例子
Here we go.This is a very good

这是一个很好的例子
example for us to understand better.

它是对称的
You can say,this is really

它是对称的
some symmetric matrix.

如果大家考虑LDU分解
When you just consider

如果大家考虑LDU分解
somehow the same procedure，

如果大家考虑LDU分解
as we did before，

将会得到一个简单的形式
like lLDU factorization，

将会得到一个简单的形式
you will be having

将会得到一个简单的形式
some special form.

L等于U的转置
You can say, L is the transpose of U.

L等于U的转置
Definitely in the future,

所以以后如果考虑的矩阵是对称的
you can say if you have a

所以以后如果考虑的矩阵是对称的
symmetric matrix，

那么L就很好确定了
you don't need to

那么L就很好确定了
get this kind of L like the

那么L就很好确定了
way we did before.

只要把U转置就可以了
Because you can simply

只要把U转置就可以了
take the transpose of this U

这是对称矩阵的特性
and you'll be able to get L,

这是对称矩阵的特性
that's the nature of this

这是对称矩阵的特性
kind of result.

最后一个性质是
No problem.

最后一个性质是
Ok. The last one is like this one

如果求这个矩阵的逆矩阵和转置
if you take the inverse of this matrix,

如果求这个矩阵的逆矩阵和转置
that means if you have an invertible

如果求这个矩阵的逆矩阵和转置
matrix and you do these

如果求这个矩阵的逆矩阵和转置
two operations.

可以求逆再转置
First one is inverse and the second one

也可以交换顺序
is transpose. Actually, you can reverse the

也可以交换顺序
order of these two operations.

也就是说先转置
You can take the transpose first,

也就是说先转置
and to talk about this inverse. Ok.

再求逆矩阵
Very much like this.

所有的同学都可以做一些
And everyone can just do

所有的同学都可以做一些
a lot of exercises

练习来理解我们讲到的
and to understand better of

这些概念和性质
this particular definitions.

好的 这节课我们就讲到这
And that's all for this lecture.

谢谢大家
Thank you very much.