Definition of eigenvalues and eigenvectors of a matrix

Let A be any square matrix. A non-zero vector v is an eigenvector of A if

          Av = λv

for some number λ, called the corresponding eigenvalue.

NOTE: The German word "eigen" roughly translates as "own" or "belonging to". Eigenvalues and eigenvectors correspond to each other for any particular matrix A. 

How to find the eigenvalues and eigenvectors of a matrix

First we consider a 2×2 matrix say A.

Now

1. Set up the characteristic equation,

 using |A − λI| = 0.

2. Solve above characteristic equation, and we get  eigenvalues (2 eigenvalues for a 2x2 system)

3. Substitute the eigenvalues into the equations given by (A − λI)v = 0 and solve. 

Here we get two eigenvectors of A for corresponding two eigen value. 

Step 2. Solve the characteristic equation, giving us the eigenvalues 

Step 3. Substitute the eigenvalues into the two equations given by (A − λI )v = 0.

Homework:- 

1. 

2.