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信息不丢矢,左逆有意义:判断逆矩阵存在的简单条件 – 译学馆
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信息不丢矢,左逆有意义:判断逆矩阵存在的简单条件

Matrix inverses make sense: a simple condition for when the inverse exists

In a linear algebra course,
在线性代数的课程中
Matrix inverses are something that get a lot of attention.
矩阵的逆是一个备受关注的知识点
There are all these formulas thrown at you
你会遇到许多这样的公式
like for the determinants, the inverses of a 2×2 matrices
比如行列式 2×2矩阵的逆
cramner’s rule etc etc.
克莱姆法则等等
[好吧 我承认我是查了当年的课本才写得出这些公式 毕竟这些知识早就已经尘封在痛苦记忆的深处了]
These formulas are great,
这些公式很重要
but I think they can obscure the very simple idea behind
但我认为其背后可能蕴藏着更简单的思想
What an inverse actually is, and when it exists.
逆矩阵究竟是什么 它在什么条件下存在
You’ve probably been taught that the inverse exists
你或许有学过 当且仅当行列式不为零时
if and only if the determinant isn’t 0.
矩阵才可逆
But most students have no idea
但是多数学生甚至都不知道
what the determinant has to do with anything
行列式究竟有什么用处
let alone inverses.
更别提它和逆矩阵的关系了

[(紫色字体)我看不明白]
I’m going to teach a completely different condition for when an inverse exists,
那现在我要讲一个在我看来更简单易懂的
that I think is much more intuitive.
完全不同的逆矩阵存在判定条件
But before all that:
在开始前我想先问问大家
Do you know what the linear transformation is?
你们知道线性变换的概念吗?
If not, go watch my last video because
如果不知道的话 请去看我的上一个视频
none of this one is going to make sense at all otherwise.
否则你可能会无法理解这节课的内容
In fact, if you want to brush up on vectors and bases,
如果你想复习一下向量和基矢的相关知识
you can watch the video before that too.
之前的视频其实也是一个不错的选择
Even though this video is about matrix inverses,
虽说这个视频的主题是矩阵的逆
I’m not going to define what an inverse is straight away.
但我不会直接去定义什么是逆矩阵
Instead I’m going to define something called a left inverse
相反我要定义一个叫“左逆矩阵”的东西
because I think understanding these first
因为我相信 对这个概念的认识
will give you a better intuition for inverses themselves.
可以帮助你更好地理解逆矩阵
Imagine you have a bunch of vectors
想象一下 在某二维向量空间中
in some vector space, in this case in 2d,
有这样一组向量
and you apply some Matrix M on them, here it rotates them.
将旋转矩阵M作用于它们
Then you think you’d like to undo
接下来你又想还原刚刚的操作
what you just did and get the vectors back to where they were.
让这些向量变回原样
In this example,
在这个例子中
what transformation undoes the rotation?
什么样的变换能还原之前的旋转操作?
you’d just rotate everything back by the same angle. Right?
只要往回转相同的角度就可以了 对吧?
That’s all a left inverse is.
这就是我们所说的左逆矩阵
It’s the matrix that undoes the original matrix,
原来的矩阵可以由这个矩阵还原得来
so it’s like you’ve done nothing at all.
也就是说 像是什么都没做一样
If I wanted to write this as an equation,
如果我想把它写成方程的形式
it’d say, if you do M,
就是说 如果使用变换矩阵M
then you do L,
再使用变换矩阵L
It’s the same as if you did nothing.
其实跟什么都不做没啥区别
This thing is called the identity matrix, by the way
顺便提下 这玩意叫单位矩阵
and it just means the transformation where you do nothing.
它表示等效于不产生任何作用的矩阵变换
This thing is called the left inverse for… hopefully obvious reasons.
此即“左逆矩阵” 希望我说得够明白了
So now we know,
现在我们知道了
the left inverse is a matrix that undoes the original matrix’s action.
左逆矩阵可以还原原变换矩阵施加的作用
The annoying thing about inverses is really their name.
逆运算的烦人之处的确在于它们的名字
It sounds like the *inverse* should
因为听起来它们就像是
be the thing that undoes a matrix.
专门用来还原矩阵的一样
Instead,the definition of M inverse is:
但其实 M逆矩阵的定义是:
M inverse undoes M
M的逆可以还原M矩阵
And M undoes M inverse.
而M矩阵又可以还原M的逆
Going back to our example:
回到我们之前的例子:
if you do M first and then L, that’s the identity.
依次做M和L变换 等于被单位矩阵作用
But it’s also true that If you did L first,then M,
而如果你是先做L变换再做M变换
you’d also get the identity.
同样也等价于单位矩阵
So since L undoes M and is undone by M,
正因为L既能还原M矩阵也可以被M还原
L and M are inverses of each other.
我们说L矩阵和M矩阵互为对方的逆
[L和M互为对方的逆!]
You might wonder, is the left inverse
你可能会想 左逆矩阵
always also the inverse like this?
是否总是也等于像这样的逆矩阵?
No, obviously not, or they wouldn’t have different names, would they?
显然不 否则他们的名字不会不同对吧?
[不能剧透!!!]
Before we move on, let me ask you a question
在我们继续上课前 让我来考考你
to check you’ve understood this so far.
看看到目前为止你是否真的理解了
Imagine you have a matrix like this.
假设有这样的一个矩阵
What it does is that it takes a 3D vector, and jumbles up the components.
它的作用是改变一个三维向量元素的排列
Does this matrix have a left inverse?
那这个矩阵有左逆吗?
As in, can you undo this?
换句话说 你能还原之前的操作吗?
Then if it does have a left inverse,
如果它有左逆矩阵的话
figure out if it has an inverse as well.
再看看它是否也有逆矩阵呢
Put you answer in the poll in the corner,
现在暂停视频去思考一下
and pause the video now to think about it.
然后告诉我们你的答案
[A矩阵有左逆吗? A矩阵有逆吗?]
The answer is that it does have a left inverse.
答案是这个矩阵确实有左逆
It’s the one that takes a vector like this
它像这样作用于一个向量
and rearranges the components like this.
又像这样重新排列组合其组成元素
It’s clear that this is a left inverse of A
很明显B是A的左逆矩阵
since it undoes it like so.
因为B以这样的方式还原了A
But A is also a left inverse of it,
但是A同时也是B的左逆矩阵
as you can see, because A undoes this matrix.
正如你所见 因为A还原了B这个矩
So B is the inverse of A.
所以B矩阵是A矩阵的逆矩阵
Now that we know what an inverse is,
现在我们已经知道什么是逆矩阵了
let’s think more about when they exist or don’t.
让我们来进一步讨论它们存在与否的条件
Again, it’s going to be more convenient to look at
同样的 先考虑什么时候存在左逆
when a left inverse exists first.
会更加方便一点
Here’s another question.
还有一个问题
This matrix takes a 2d vector a b and sends it to a 0.
这个矩阵把二维向量a,b变换成a,0
Does this matrix have a left inverse?
那它有左逆矩阵吗?
If so, figure out what it is.
如果有的话 想想它会是什么
Again put your answer
暂停视频思考一下
in the poll in the corner and pause now to think about it.
再来告诉我们你的答案
Notice something about this matrix.
注意一下这个矩阵
It takes the vector a b to a 0,
它可以把向量a,b变换成a,0
but it also takes a d to a 0 as well.
但它同样也能把a,d变换成a,0
This… is a bad thing,
这可不是个好消息
and it’s because of this that the left inverse doesn’t exist. Why?
也正因为此 它才没有左逆矩阵 为何?
Well, say you have some vector V and M takes it to W.
假设你用矩阵M把某向量V变换为W
The left inverse of M, if it exists, knows M and what W is,
如果M的左逆存在 又已知M和W
but it doesn’t know what the vector that produced W was.
但我们却不知道产生W的原有向量是什么
Just using the information given, it needs to find what the original vector was,
需要通过已知的信息来找到原向量
so that it can take W back to where it came from.
这样才可以把向量W还原为原向量V
However. If there’s some other
然而 假如存在另一个
vector U that also goes to W,
同样也能变换到W的向量U
the left inverse has a problem.
那这个左逆矩阵就有问题了
it can’t just look at W and know for sure it whether
因为信息不足 仅凭W我们不能确定
it came from V or from U because there isn’t enough information.
它到底是来自于V还是U
This means the left inverse
这意味着左逆矩阵
*can’t* take w back to where it came from, so… it doesn’t exist.
“不能”对W进行还原 所以左逆不存在
This thing here,
这里是说
where two different vectors V and U
当两个不同的向量V和U
get mapped to the same vector,
映射到同一个向量时
i.e M(U)=M(V),
即M(U)=M(V)
is what I’ll call M losing information.
也就是我们所说的M矩阵丢失了信息
What we’ve just seen is that if M loses information
可以看到 假如M丢失了信息
it doesn’t have a left inverse.
它就没有左逆矩阵
But what about the other way around?
但是如果反过来呢?
[M矩阵无信息丢失]
If M doesn’t lose information
如果M没有丢失信息
does this mean the left inverse exists?
这意味着左逆存在吗?
Well, yes actually.
确实是这样的
Because all the left inverse has to do
因为左逆矩阵所需做的
to undo M is find the vector W came from.
就是找到W的原向量来还原M矩阵的作用
Since there’s only one vector V it could be,
因为只能有一个向量V
there is an inverse that takes W and returns V.
所以一定存在可以使W变回V的逆矩阵
This doesn’t mean it’s easy to find out what V is necessarily,
不过这并不意味着V很容易求
but looking at W does in principle give you enough information
但理论上仅凭W确实能够获得足够的信息
to undo M and return V.
来还原M矩阵的作用并得到原向量V
So a matrix has a left inverse if and only if
所以当且仅当一个矩阵不发生信息丢失时
it doesn’t lose information.
它才有自己的左逆矩阵
[信息不丢矢 左逆有意义]
Let’s look at another example
让我们再看一个例子
to understand this point better.
来更好地理解这一点
Imagine I have a matrix from 2d to 3d
假如我有这样一个从二维到三维的矩阵
and what it does is,
它的作用在于
it rotates any 2D vector into 3D space like this.
将任意的二维向量像这样旋转到三维空间
[不好意思,再来一次]
[我真是个小机灵]
Does this matrix have a left inverse?
那这个矩阵有左逆吗?
Pause the video to think about it.
大家暂停视频思考一下
The answer is, it does have a left inverse
答案是它确实有左逆矩阵
because A doesn’t lose information.
原因是A没有发生信息丢失
If you want to take vectors like this back,
如果你想把这样的向量还原回去
you know where they came from
又已经知道它是怎么变换过来的
so all you have to do is rotate the plane back.
那你要做的就是把这个面转回原来的位置
Let B be a matrix that
假设B是这样一个矩阵
takes 3D vectors to 2D that does rotate this plane back.
它能通过转回这个面来将三维变换到二维
It is a left inverse of A.
那它就是A的左逆矩阵
Now, is B the inverse of A?
那么问题来了 B是A的逆矩阵吗?
In other words, is A B’s left inverse?
换句话说 A是B的左逆矩阵吗?
Pause the video and think about it.
暂停视频好好想一下
The answer is, no, B has no left inverse
答案是 不 B没有左逆矩阵
We’ll show that by showing B loses information.
我们将通过证明B会丢失信息来加以论证
First, pick any 3D vector that’s not on this plane
首先 选择一组不在这个面上的三维向量
[不是这个面]
B has to send it to some 2D vector,
B矩阵要将它变换成二维向量
so let’s just say it’s here.
咱们假设它在这儿
But there’s another 3D vector that’s been already sent there.
但是已经有另一个三维向量在那里了
It’s this vector u that’s on the plane.
它是在这个平面上的向量u
So B(u) is equal to B(v).
因此B(u)等于B(v)
Hence B loses information and doesn’t have a left inverse.
也即B丢失了信息 所以没有左逆
There’s an important lesson to be drawn from this example.
从上述例子中我们学到了一个重要的知识
You might have wondered
你可能已经在想
why we only ever talk about the inverses of a square matrices.
为什么我们只讨论了方矩阵的逆
What’s so special about transformations from n dimensions to n dimensions?
从n维到n维的矩阵变换有什么特别的?
The reason is, non square matrices, i.e.
原因在于非方阵 也就是
ones from n dimension to m dimensions never have an inverse.
n维到m维的变换矩阵永远不会有逆矩阵
The issue is, if you have any transformation going from a bigger space to a smaller space,
事实上如果进行了空间上从大到小的变换
like B which went from 3d to 2d,
比如从三维到二维变换的B矩阵
you have to lose information.
那就一定丢失了信息
These types of matrices always send some vectors
这种类型的矩阵总能把多个不同的原向量
to the same place.
对应变换为一个
If you have a matrix from a smaller to a bigger dimension,
如果你有一个从低维到高维变换的矩阵
it’s possible that it has a left inverse
它就有可能有左逆
like A did in our example.
像我们例子中的A矩阵那样
But it’s left inverse goes from big to small, like B,
但它的左逆是B那样从高维到低维的矩阵
and so it can’t be undone.
所以是无法被还原的
Hence, even though some nonsquare matrices have left inverses,
因此 即使有些非方阵矩阵有自己的左逆
they never have an inverse.
它们也一定没有逆矩阵
Square matrices don’t have these issues at all though.
而方阵就根本不会出现这些问题
Actually, for square matrices, everything massively simplifies
事实上 对方阵来说一切都大大简化了
because the left inverse is always equal to the inverse.
因为方阵的左逆矩阵总是它的逆矩阵
So if a square matrix has a left inverse,
因此 如果一个方阵有自己的左逆
it automatically has an inverse.
它自然就有一个逆矩阵
I am not going to prove this fact.
我并不打算证明它
You are, for homework.
不过你可以把它当成课后练习
But I will give you an illustrative exanple in a little
但接下来我会给你们一个例子
to help you understand why it’s true.
来帮助你理解为什么它是正确的
[对于方阵 左逆即逆 我叫它“例证法” 但我的线代老师却不屑一顾]
Once you’ve proved it, you’ ll see that for a square matrix A,
证明后你会发现 对于方阵A而言
if B undoes A,
如果方阵B能还原A
then A undoes B as well.
那么反过来方阵A也能还原B
This gives us an easy criteria
这为我们检查A是否有逆矩阵
for checking whether A has an inverse or not,
提供了一个简单的方法
because it’s the same criteria we used to check
因为这和我们之前用来检验
whether A has a left inverse:
A是否有左逆矩阵的办法相同:
Just ask, does A lose information.
只需要考虑A会不会丢失信息就可以了
If yes… then sorry, A inverse doesn’t exist.
如果会的话…那很遗憾A的逆就不存在
If no, then A inverse does exist.
如果不会 则说明A矩阵的逆存在
You might be thinking, ok, so what?
你可能在想 好吧 不过那又怎样?
How is this easy to check?
为啥说这样检验就容易了呢?
Wouldn’t you have to compute the outcome for every single vector
难道不需要对每个与M对应的向量
that goes into M and compare the results to every other vector
进行计算 再比较不同向量对应的结果
and see if any of them match?
看看结果是否相互匹配?
Isn’t that beyond tedious?
那岂不是很麻烦吗?
Thankfully there is an easy way to check this condition.
幸运的是 我们有一种简单的办法来检验
All you have to do is figure out
你需要做的只是弄清
which vectors get mapped to 0.
哪些向量会映射到零向量
For any linear map,
对任何线性映射来说
0 is always mapped to 0,
零向量总是映射到零向量
but all you need to do is find out
但是你需要做的仅仅是检查
if there’s any *other* vector mapped to 0 or not,
是否有其他向量映射到零向量
and that’s enough to decide if M loses information.
这样就足以判断M是否有丢失信息
Why would that be enough?
但为什么这就足够了?
Imagine two vectors U and W do both go to W,
假设两个向量U和V都映射到W
so M loses information.
所以M就丢失了信息
Then the vector U-V, by linearity, gets mapped to 0
又由线性关系知向量U-V映射到零向量
So whenever you have 2 vectors going to the same thing like this,
所以每当两个相同的向量映射对象相同时
you always get at least 2 vectors ending up at 0.
就总存在至少两个能映射到零向量的向量
So you can check whether a matrix loses information
因此我们可以通过检查有多少向量
by just checking how many things go to zero.
映射到零向量来判断矩阵是否丢失了信息
In other words, figure out how many vectors V
换句话说就是要弄清楚 有多少向量V
satisfy the equation MV=0.
满足方程MV=0
This is part of why you spend so much time
这就是为什么你要花这么多时间
in a linear algebra course studying the solutions to equations like this.
在线性代数课程中学习解这些方程的原因之一
You can solve for V by a) using subsitution
你可以通过以下方法来求解V 换元法
b) using Gaussian elimination
高斯消元法
or (c) by getting your computer to do the Gaussian elimination for you.
或者用计算机来做高斯消元
But the point is, you can find out if M loses information easily enough this way
重点在于这样你就能得知M是否丢失信息
and that tells you whether M has an inverse.
从而进一步判断M有没有逆矩阵
Let me summarize quickly
让我来简要总结一下
what we learnt in this video:
我们从这个视频中学到了什么
[因为这个视频有点儿长 既对你们 也对我们]
1. A left inverse of a matrix is matrix that undoes it.
一 矩阵的左逆矩阵还原该矩阵的作用
2. That the left inverse exists if you don’t lose information:
二 如果矩阵没有丢失信息 左逆就存在
i.e, if the matrix never sends two different vectors to the same vector.
即不会把两个不同向量变换成同一个向量
3. The inverse is the matrix that both
三 逆矩阵还原原矩阵的变换作用
undoes the matrix, and is undone by the matrix
而自身的变换作用也被原矩阵还原
4. Nonsquare matrices never have inverses.
四 非方阵的矩阵一定没有逆
5.For a square matrix,
五 对于方阵而言
the left inverse is equal to the inverse
其左逆矩阵等价于逆矩阵
6. You only need to check if a square matrix
六 就方阵而言 只需要检验它
loses information or not to decide if it has an inverse.
会不会导致信息丢失 就能判断是否有逆
7. You can check whether the matrix loses
七 通过观察有多少向量映射到零向量
information by looking at how many different vectors get mapped to 0.
来判断矩阵会不会出现信息丢失
This you can do by solving the equation Mv=0.
你可以通过解方程Mv=0来进行判断
And so that’s it. But before you run off,
这节课就上到这里 但先不要急着关视频
here’s some homework for you.
我们还为大家准备了一些课后作业
The first one is multiple choice.
首先是一个多选题
Which of these is the inverse of this matrix?
选项中哪些是题中所给矩阵的逆矩阵?
I know that you can just check
我知道你可以直接代入计算
each of these to see which works,
来检验哪个选项是可行解
but I’d rather you did it another way.
但我更希望你采用另一种方式
[就当这种取巧的方法配不上俺们这些好学生吧]
And for crying out loud,
为了进一步掌握今天的知识
don’t use the formula for the inverse.
请不要使用求逆的公式
Once you’ve figured it out, put your answer in the poll.
算完的同学把你们得出的答案发给我们吧
Question 2. Prove that for a square matrix
第二个问题 证明对于方阵M
M inverse is equal to the left inverse of M.
其逆矩阵等价于其左逆矩阵
There’s lots of hints in the description for this,
在这题的描述中就有很多提示
but first I want you to try question 3
但我想让你先看看第三问
because it’s a very illustrative example.
因为这个例子很能说明问题
[当我说我来做例题演示的时候 其实是要你来做:)]
Question 3. Suppose we have the matrix from before.
问题三 假设还是上题那个矩阵
If we have a left inverse for it, L,
如果它有一个左逆矩阵L
then we know L undoes M.
现在我们知道L能还原M做的变换
We also know M takes the basis vectors to these vectors,
而M又能把基矢变换成这些向量
so L must take them back.
所以L就一定能把它们变回去
First show that these two new vectors form a basis.
首先证明这两个新向量可以构成一组基矢
Then, to show that M undoes L,
然后 为了证明M确实能够还原L的作用
we need to show that for any vector v, if you apply L
需要证明 对任意向量v 如果先左乘L
and then M, you get v back.
再左乘M 你就能得到原来的v
Show this by writing v as
通过把v向量写成第一部分提到的
a linear combination of the basis in the first part.
基向量线性组合的形式 就能证明这一点
Hopefully doing this first will help you with the proof in question 2.
希望先这么做能对问题二的证明有所帮助
As you will have noticed, the first question was from Brilliant.org.
你可能已经注意到 问题一来自网站Brilliant. org
What I like about questions from there it is
我比较喜欢那个网站的问题
that they don’t just give you a formula and then ask questions
因为他们不会只是给你公式再以此出题
where you plug numbers into that formula.
那样你就会简单地把数字代入公式
That’s what I found a lot of highschool and early university textbooks do,
这种情况出现在高中以及早期的大学课本上
and it’s annoying because that doesn’t teach you anything.
可恶的是这样并没有真正教会你任何东西
Instead, they get you to do examples like this one,
相反 这个网站会通过给你做例题
and then understand the principle yourself,
来让你自己理解其中的原理
then in the next few questions,
然后再在接下来的几个问题中
lead you to finding the general solution on your own
引导你自行找到一个普适的解决方法
As you can see,
如你所见
this is exactly how I like to learn new maths
这正是我喜欢的学习新的数学知识的方式
before reading the proof of anything I’ll do lots and lots of simple examples first
在学习具体证明前我先会做大量简单练习
and try to understand the underlying reasoning
来试着理解问题背后最基本的道理
so I love how Brilliant allows you to do this in a structured way.
我喜欢这个网站所提供的结构性学习方式
They have loads of different maths and science course on their website,
他们的‎网站上有许多数学和科学课程
which you can access completely with a monthly or yearly membership.
如果你有月卡或者年卡 就可以随意访问
If you follow the link in the description
如果你点击介绍中的或屏幕上的链接
or on screen, you can get 20% off an annual subscription.
或者在屏幕上 你可以免除20%的年费。好的
Alright, so that’s all for this video,
好啦 这就是本期视频的全部内容
I hope you enjoyed it.
希望你们喜欢
The next one is about changing basis,
下一个视频与“基矢变换”有关
which is key to understanding loads of Quantum mechanics,
这是理解包括海森伯测不准原理在内
including the Heisenberg uncertainty principle.
很多量子力学知识的关键
It should be up here in about 2 weeks,
只要我不像对这个视频的最初版本那样
as long as I don’t dislike I hate it just after uploading it
在上传之后变得寝食不安悔恨交加
like I did with the original version of this one.
那新的视频会在两周后和大家见面
Anyway, subscribe if you’d like to be notified
总之 如果你想及时看到下一个视频
when my next one is up. Thanks for watching!
就请关注我吧 谢谢观看!

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视频概述

定义左逆矩阵,进而介绍一个判断逆矩阵是否存在的方法。

听录译者

收集自网络

翻译译者

唐宋元明清

审核员

审核员1024

视频来源

https://www.youtube.com/watch?v=ESKcF8XFzLM

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