Find the eigen values and eigen vectors of A = [[1, 1, 3], [1, 5, 1], [3, 1, 1]]

Step 1: Find the characteristic equation

The characteristic equation of a matrix A is given by det(A - λI) = 0, where I is the identity matrix and λ are the eigenvalues. For the given matrix A, we have:

A - λI = [[1-λ, 1, 3], [1, 5-λ, 1], [3, 1, 1-λ]]

So, det(A - λI) = (1-λ)[(5-λ)(1-λ) - 1] - 1[(1-λ) - 3] + 3[1 - (5-λ)]

Solving this gives the characteristic equation as λ^3 - 7λ^2 + 9λ - 5 = 0.

Step 2: Find the eigenvalues

The eigenvalues are the roots of the characteristic equation. Solving λ^3 - 7λ^2 + 9λ - 5 = 0 gives the eigenvalues as λ1 = 1, λ2 = 2, λ3 = 4.

Step 3: Find the eigenvectors

The eigenvectors are found by substituting each eigenvalue back into the equation (A - λI)v = 0 and solving for v.

For λ1 = 1, we get the system of equations:

0v1 + v2 + 3v3 = 0 v1 + 4v2 + v3 = 0 3v1 + v2 + 0v3 = 0

Solving this system gives the eigenvector v1 = [1, -1, 1].

For λ2 = 2, we get the system of equations:

-v1 + v2 + 3v3 = 0 v1 + 3v2 + v3 = 0 3v1 + v2 - v3 = 0

Solving this system gives the eigenvector v2 = [1, 1, 1].

For λ3 = 4, we get the system of equations:

-3v1 + v2 + 3v3 = 0 v1 + v2 + v3 = 0 3v1 + v2 - 3v3 = 0

Solving this system gives the eigenvector v3 = [1, -2, 1].

So, the eigenvalues are 1, 2, 4 and the corresponding eigenvectors are [1, -1, 1], [1, 1, 1], [1, -2, 1].