Detailed calculation process of matrix eigenvalues and eigenvectors

1. Definition of matrix eigenvalues ​​and eigenvectors

        A is an n-order matrix. If the number λ and the n-dimensional non-zero column vector x satisfy Ax=λx, then the number λ is called the eigenvalue of A, and x is called the eigenvector of A corresponding to the eigenvalue λ . ModeAx=λx can also be written as (A-λE)x=0, and |λE-A| is called the characteristic polynomial of A. When the characteristic polynomial is equal to 0, it is called the characteristic equation of A. The characteristic equation is a homogeneous system of linear equations. The process of solving the characteristic value is actually solving the solution of the characteristic equation.

 

Compute: eigenvalues ​​and eigenvectors of A.

Calculate the determinant

Simplified to:

Get eigenvalues:

Simplified to:

 

Let get the feature matrix:

Similarly, when :

，

Let get the feature matrix: