Matrices Template

Linear transformation

Consider the matrix A=[[2,1],[1,2]]

This matrix defines a linear transformation x↦A*x for vectors x∈ℝ^2.

Applying the transformation

Let x=[[1],[0]]

Compute A*x

A*x=[[2,1],[1,2]]*[[1],[0]]=[[2],[1]]

Repeated application

Apply the transformation again:

A^2*x

First compute

A^2=A*A

A^2=[[2,1],[1,2]]*[[2,1],[1,2]]=[[5,4],[4,5]]

Eigenvalues

To understand the transformation, compute the eigenvalues.

det(A−λ*I)=0

|[2−λ,1],[1,2−λ]|=(2−λ)^2−1=λ^2−4*λ+3

Result

(λ_1)=3, (λ_2)=1

Eigenvectors

For (λ_1)=3

(A-3*I)*v=0

[[−1,1],[1,−1]]*v=0

One eigenvector is (v_1)=[[1],[1]]

For (λ_2)=1

(v_2)=[[1],[−1]]