Lecture 10 Eigenvalue and Eigenvector

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Textbook:  Introduction to Linear Algebra, Fifth Edition (2016)

Textbook open resource: http://math.mit.edu/~gs/linearalgebra/

0. 引言

這是第二部分的第一章。在第一部分中，我詳細地討論了線性方程的求解，解的條件，以及解的空間，最終得到描述一個空間最精煉的語言 - 基（basis）。現在，我將把目光更多地聚焦在線性空間上：如何描述一個線性空間？線性空間的特徵是什麼？有什麼特殊的線性空間？它們的性質是什麼樣的？最終我將充分討論什麼是線性變換，以及其代數實質。歡迎來到新大陸！

（本文不適合暗黑模式下食用哦！推薦橫屏閱讀。）

1. Eigenvalues and Eigenvectors of Square Matrix

Consider a square matrix

Definition 1.1 The basic equation is

If

It is important that the eigenvalues of matrix are not necessary to be real numbers. In other words, eigen-system should be in

To solve the eigenvalues, it is to solve this linear equation

It is to find the null space of matrix

This is equation for the eigenvalues, which is a polynomial equation. (Very Important!) Eigenvalues are the roots of this polynomial. Once you obtain the eigenvalues, solve

Example 1.1: Solve the eigen-system of rotation matrix

Find the roots:

Examine equation for the eigenvalues. It is actually the mono-variable function of fundamental theorem of algebra,

By definition:

Therefore:

A direct corollary is that a matrix with zero eigenvalues is singular.

On the other hand, Vieta formula tells us:

What is

It is called trace.

Next, the eigenvalues of matrice related to

The eigenvalue of

The eigenvalue of

Eigenvalue of

2. Cayley-Hamilton Theorem

Arthur Cayley, a briliant British mathematician who worked mostly on algebra, wrote in his masterpiece A memoir on the Theory of Matrices:

I obtain the remarkable theorem that any matrix whatever statisfies an algebraic equation of its own order, the coefficient of the highest power being unity... and the last coefficient being in fact the determinant.

He wanted to say that for any matrix

Definition 2.1: If

Since

Multiplication with constant

From the perspective of linear combination,

should be linear dependent because there are more . Or there always exists a coefficient set

So it is not surprising the existence of a polynomial that causes

Theorem 2.2 (Cayley-Hamilton Theorem): If

To prove this theorem, first introduce the concept of cofactor matrix. Consider an invertible matrix

Also, you can use confactors to calculate the inverse of

You may ask why

Since right side is which , the inverse of

Let's prove Cayley-Hamilton theorem.

Proof: Suppose

By definition of

Plug into LHS:

And compare to RHS:

Here is the most intriguing part:

Top up two sides:

Complete.

3. Exercises

Exercise 1.1: A 3 by 3 matrix

Exercise 1.2: Find the eigenvalues of this permutation matrix

Exercise 2.1: Square matrix

4. Challenge Problems

Calculate the power of matrix

Hint: Use Calyley-Hamilton theorem. Notice that