MIT18.06_7_Application of eigen

Diagonalization and powers of

We know how to find eigenvalues and eigenvectors. In this lecture we learn to diagonalize any matrix that hasindependent eigenvectors and see how diagonalization simplifies calculaions. The lecture concludes by using eigenvalues and eigenvectors to solve difference equations.

我们已经了解，如何找到特征值和特征向量了。
在这一节中，我们学会对角化任意包含个独立特征向量的矩阵，以及
利用对角化来简化计算。

Digagonalizing a matrix

对角化矩阵

Ifhaslinearly independent eigenvectors, we can put those vectors in the columns of a (square, invertible) matrix. Then

对角化的一个重要前提，矩阵的个特征向量线性独立(这样矩阵可逆。
这些特征向量组成特征向量矩阵。

上式是 key equation 的代入。

上一步涉及 列变换

Note thatis a diagonal matrix whose non-zero entries are the eigenvalues of. Because the columns ofare independent,exists and we can multiply both sides ofby:

Equivalently,.

这种方法的一个重要前提：矩阵有个独立的特征向量，即满秩，可逆

如果矩阵的所有特征值都不同，那么矩阵拥有个独立特征向量（可以实现对角化

Powers

What are the eigenvalues and eigenvectors of?

• If,
• then.

如何求得的特征值和特征向量?

The eigenvalues ofare the squares of the eigenvalues of.

的特征值是对应的特征值的平方。

The eigenvectors ofare the same as the eigenvectors of.

的特征向量与的特征向量相同。

If we writethen:

Similarly,tells us that raising the eigenvalues ofto thepower gives us the eigenvalues of, and that the eigenvectors ofare the same as those of.

矩阵是由 特征向量 组成。
的表达形式，可以看出：

• 的主对角线元素为特征值的阶。
• 的形式反映，是特征向量组成方阵。

Theorem: Ifhasindependent eigenveectors with eigenvalues, thenasif and only if all.

定理： 当时，当且仅当所有.

is guaranteed to haveindependent eigenvectors (and be diagonalizable) if all its eigenvalues are different.
Most matrices do have distinct eigenvalues.

• 仅当矩阵中的所有特征值都不同，才可保证有个独立特征值。
• 绝大多数的矩阵都有不同的特征值。

Repeated eigenvalues

Ifhas repeated eigenvalues, it may or may not haveindependent eigenvectors.

若矩阵有重复的特征值，可能但不一定有个独立的特征值。

For example, the eigenvalues of the identity matrix are all 1, but that matrix still hasindependent eigenvectors.

单位矩阵所有特征值都为1，有n个不同的特征向量。

Ifis the triangular matrixits eigenvalues are 2 and 2. Its eigenvectors are in the nullspace ofwhich is spanned by. This particulardoes not have two independent eigenvectors.

• 单位矩阵含个特征向量（也是单位矩阵
• 矩阵就只有一个特征向量。

Difference equations

差分方程

Start with a given vector. We can create a sequence of vectors in which each new vector istimes the previous vector:.is a first order difference equation, andis a solution to this system.
We get a more satisfying solution if we writeas a combinating of eigenvectors of:

Then:

and:

这里涉及特征值，方程上的应用。

Fibonacci sequence

斐波那契数列

The Fibonacci sequence is 0,1,1,2,3,5,8,13,… In general,.

斐波那契的定义

If we could understand this in terms of matrices, the eigenvalues of the matrices would tell us how fast the numbers in the sequence are increasing.

如果我们能从矩阵形式的角度，理解Fibonacci序列。
将很容易理解，Fibonacci序列的增长速度。

was a first order system.is a second order scalar equation, but we can convert it to first order lienar system by using a clever trick. If, then:

is equivalent to the first order system.

用矩阵形式表示，斐波那契数列的关系式，就得到.

What are the eigenvalues and eigenvectors of?
Becauseis symmetric, its eigenvalues will be real and its eigenvectors will be orthogonal.

矩阵是对称矩阵，故特征值为实数，并且特征向量正交。

Becauseis a two by two matrix we know its eigenvalues sum to 1 (the trace) and their product is -1 (the determinant).

• two by two matrix 特征值之和为1 (equal to trace)
• 特征值之积 (equal to det A)

Setting this to zero we find; i.e.and.

可以直接算出特征值。

The growth rate of theis controlled by, the only eigenvalue with absolute value greater than 1.

的增长速率仅仅与相关

This tells us that for large,for some constant.
(Remember, and heregoes to zero since).

是对等式：的分解表达。

To find the eigenvectors ofnote that:

equalswhen, soand.

Finally,tells us that. Because, we get:

Using eigenvalues and eigenvectors, we have found a closed form expression for the Fibonacci numbers.

实用特征值和特征向量我们可以求得Fibonacci数列的闭合表达式

Summary:
When a sequence evolves over time according to the rules of a first order system, the eigenvalues of the matrix of that system determine the long term behavior of the series.
To get an exact formula for the series we find the eigenvectors of the matrix and then solve for the coefficients

当一个序列按照一阶系统的规则随时间演化时，该系统矩阵的特征值决定了该序列的长期行为。

为了得到这个级数的精确公式，我们先找到矩阵的特征向量，然后解出系数

Differential equations and

The system of equations below describes how the values of variablesandaffect each other over time:

上面等式系统，描述了变量和与他们关于时间的相互影响。

Just as we applied linear algebra to solve a difference equation, we can use it to solve this differential equation.
For example, the intial conditioncan be written。

就如我们应用线性代数解决差分方程一样。我们利用线性代数来解决微分方程问题。
这里举一个例子，向量的初始值为为这个向量的初始值。

Differential equations

By looking at the equations above, we might guess that over timeswill decrease.

通过观察以上等式，我们也许可以猜测，随着时间发展会衰减。

We can get the same sort of information more safely by looking at the eigenvalues of the matrixof our system.

我们可以通过观察特征值矩阵来更保险地了解到这相同的信息。

Becauseis singular and its trace is -3 we know that its eigenvalues areand.
The solution will turn out to includeand. Asincreases,vanishes andremains constant.

矩阵的trace为-3，我们可以计算得到和。
解的组成包含元素和。
随着t的增大，趋近于零，是零。

Eigenvalues equal to zero have eigenvectors that are steady state solutions.

特征值为零，对应的特征向量是稳定解。

is an eigenvector for which.
To find an eigenvector corresponding towe solve:

and we can check that. The general solution to this system of differential equations will be:

对于微分方程解来说，特征值分量的形式为：
可以利用key equation来得到闭合解。

Isreally a solution to? To find out, plug in:

which agrees with:

The two “pure” termsandare analogous to the termswe saw in the solutionto the difference equation.

对于阶系统，我们依然可以用类似于来解决问题。

Plugging in the values of the eigenvectors, we get:

We know, so at:

and.

在确定 特征值 和 特征向量 后，通解形式只需要确定和就能待定通解。当时，可以待定得到最后的通解。

This tells us that the system starts withandbut that asapproaches infinity,decays to 2/3 andincreases to 1/3. This might describe stuff moving fromto.
The steady state of this system is.

我们可以知道，当，向量.

Stability

Not all systems have a steady state. The eigenvalues ofwill tell us what sort of solutions to expert:

1. Stability:when.
2. Steady state: One eigenvalue is 0 and all other eigenvalues have negative real part.
3. Blow up: iffor any eigenvalue.
   If a two by two matrixhas two eigenvalues with negative real part, its traceis negative. The converse is not true:has a positive determinant and negative trace then the corresponding solutions must be stable.

不是所有系统，都有稳定解的。特征值会反映解的收敛情况：

1. 稳定性:when.
2. 稳定态: 一个特征值为0，其余都小于零
3. 发散: 存在

Applying

The final step of our solution to the systemwas to solve:

In matrix form:

or, whereis the eigenvector matrix. The components ofdetermine the contribution from each pure exponential solution, based on the initial conditions of the system.

在用特征矩阵表征时，会涉及到矩阵的线性组合来表示初始值，有表达式,的组成元素（components）反映了对应纯指数解(pure expoential solution)，与系统的初化值有关。

In the equation, the matrixcouples the pure solutions. We set, whereis the matrix of eigenvectors of, to get:

or:

This diagonalizes the system:. The general solution is then:

原来的等式：中，在关系式中，相互耦合。为了解耦就是实现对角化。

设置可以得到.

解耦合相关证明在下一section

Matrix exponential

含有矩阵的指数

What doesmean ifis a matrix? We know that for a real number,

We can use the same formula to define:

解耦合的表达式，看起来一头雾水。
** 其实是，泰勒展开式的矩阵形式 **

Similarly, if the eigenvalues ofare small, we can use the geometric seriesto estimate.

同理，
** 存在几何级数的矩阵形式 **

We’ve said that. Ifhasindependent eigenvectors we can prove this from the definition ofby using the formula:

It’s impractical to add up infinitely many matrices. Fortunately, there is an easier way to compute. Remember that:

能够对角化的矩阵，都可以表述为上下两式子。

When we plug this in to our formula forwe find that:

This is another way to see this relationship between the stability ofand the eigenvalues of.

Second order

We can change the second order equationinto a two by two first order system using a method similar to the one we used to find a formula for the Fibonacci numbers.

我们可以将二阶微分方程转化为矩阵形式的一阶方程
就同我们处理Fibonacci数列一样。

If, then

We could use the methods we just learned to solve this system, and that would give us a solution to the second order scalar equation we started with.
If we start with aorder equation we get abymatrix with coefficients of the equation in the first row and 1’s on a diagonal below that; the rest of the entries are 0.

若我们在处理阶方程，我们会得到一个矩阵，仅在第一行和主对角线的下一对角线有非零元素。

Markov matrices; Fourier series

In this lecture we look at Markov matrices and Fourier series - tow applications of eigenvalues and projections.

在这一节中，我们将介绍马尔可夫矩阵和傅立叶级数–两个关于特征值和投影的应用。

Eigenvlaues of

The eigenvalues ofand the eigenvalues ofare the same:

so property 10 of determinants tells us that. Ifis an eigenvalue ofthenandis also an eigenvalue of.

可以证明，矩阵和矩阵的特征值相同。

1. ,
2. 上等式两边取det
3. 等式左边行列式为0对应，同样可以令等式右边行列式为零。

Markov matrices

马尔可夫矩阵的提出跟概率思想有关。

A matrix like:

in which all entries are non-negative and each column adds to 1 is called a Markov matrix.
These requirements come from Markov matrices’s use in probability. Squaring or raising a Markov martix to a power gives us another Markov matrix.

1. 所有元素都是非负数的。
2. 每列元素总和为1。

** 马尔可夫矩阵的任意阶乘仍然满足马尔可夫矩阵性质 **

When dealing with systems of differential equations, eigenvectors with the eigenvalue 0 represented steady states. Here we’re dealing with powers of matrices and get a steady state whenis an eigenvalue.

• 面对微分方程系统问题，特征值是否为0代表着状态稳定。
• 对于幂矩阵系统问题，特征值是否为1决定状态稳定。

The constraint that the columns add to 1 guarantees that 1 is an eigenvalue. All other eigenvalues will be less than 1.

稳态条件：

1. 是一个特征值(一定存在)。
2. 其余特征值。

Remember that (ifhasindependent eigenvectors) the solution tois.
Ifand all others eigenvalues are less than one the system approaches the steady state. This is thecomponent of.

is.

Why does the fact that the columns sum to 1 guarantee that 1 is an eigenvalue? if 1 is an eigenvalue of, then:

** 如何证明特征值一定存在 **

should be singular. Since we’ve subtracted from each diagonal entry, the sum of the entries in each column ofis zero.
But then the sum of the rows ofmust be the zero row, and sois singular.
The eigenvectoris in the nullspace ofand has eigenvalue 1. It’s not very hard to find.

• Markov matrices，各列元素之和为1，主对角线各元素减去1后，各列元素和为零。（证明行向量线性相关）
• 对应的特征向量是零空间上的解，所以矩阵列向量线性相关。

We’re studying the equationwhereis a Markov matrix.

我们研究 矩阵属Markov matrix的问题。

For examplemight be the population of (number of people in) Massachusetts andmight be the population of California.
might describe what fraction of the population moves from state to state, or the probability of a single person always be positive.
We want to account for all the people in our model, so the columns ofadd to 1 = 100%.

举个例子：表征麻省人口，表征加州人口。
和组成人口列向量

矩阵描述了从一个州转移到另一个州的人口分数。

** 这里可以视为矩阵为其列向量与的线性组合 **

For example:

For the next few values of, the Massachusetts population will decrease and the California population will increase while the total population remains constant at 1000.

To understand the long term behavior of this system we’ll need the eigenvectors and eigenvalues of. We know that one eigenvalue is. Because the traceis the sum of the eigenvalues, we see that.

计算矩阵的行列式时，可以利用矩阵迹trace和行列式来简化特征值的运算。

Next we calculate the eigenvectors:

so let
From what we learned about difference equations we know that:

Whenwe have:

soand.

以上就是，利用特征值分解的全过程。

从3b1b那边看来的理解：

** 特征值是，在特征向量方向，线性变化后，向量不改变方向，仅改变角度的解。 **
我们将向量分解为，各个特征向量，在线性变化的过程中放缩，再合成，就是这里的思路。

In some applications Markov matrices are defined differently - their rows add to 1 rather than their columns. In this case, the calculations are the transpose of everything we’ve done here.

在有些应用中，Markov matrices会被定义为行和为1，使用时注意专置。

Fourier series and projections

这一节，从正交投影到傅立叶级数/

Expansion with an orthonormal basis

If we have an orthonormal basisthen we can write any vectoras, where:

Sinceunless, this equation gives.

有一组正交基，我们可以很方便地，在正交基的张成空间内表示任意向量。
同时利用正交性质，可以很方便的获取向量在各个分量上的投影。

In terms of matrices,, or. So. Because theform an orthonormal basis,and. This is another way to see that.

从矩阵乘法的角度看待可以从另一个角度的得到。

Fourier series

The key idea above was that the basis of vectorswas othonormal. Fourier series are built on this idea. We can describe a functionin terms of trigonometric functions:

This Fourier series is an infinite sum and the previous example was finite, but the two are related by the fact that the cosines and sines in the Fourier series are orthogonal.
We’re now working in an infinite dimensional vector space. The vectors in this space are functions and the (orthogonal) basis vectors are 1,,,,,

基于以上观点，外化可得：从无穷维度的函数空间，抽取相互正交的正弦函数，由它们的线性组合，表征函数。

原因：cos函数和sin函数是正交的。

What does “orthogonal” mean in this context ? How do we compute a dot product or inner product in this vector space? For vectors inthe inner product is.
Functions are described by a continuum of valuesrather than by a discrete collection of components. The best parallel to the vector dot product is:

设计函数的正交点基形式的一个特点：让函数相同的基点积后为1，正交的向量点积后为0。
这里就设计了类似相关函数的形式。

We intergrate fromtobecause Fourier series are periodic:

The inner product of two basis vectors is zero, as desired. For example,

How do we findto find the coordinates or Fourier coefficients of a function in this space? The constant termis the average value of the function. Because we’re working with an orthonormal basis, we can use the inner product to find the coefficients.

We conclude that. We can use the same technique to find any of the values.

可能是Euler或Fourier提出的，由于三角函数具有的周期性，正交（相关）积分被设置为在区间上。