Linear Algebra Question Find a diagonal matrix D and a matri

Linear Algebra Question:

Find a diagonal matrix D and a matrix P such that each of the diagonal entries of D are an eigenvalue of A and the columns of P are the eigenvectors of A. Find P^-1 and show that A=PDP^-1. Use this representation of A to evaluate A¹⁰.

Solution

As A is upper triangular, its eigen values are the diagonal elements 2,1,2 and-1.

Solving the equations Av = mv, with m any of these eigenvalues , we obtain the corresponding eigenvectors.

An eigenvector for 1 is [-1 1 0 0]\'

An eigenvector for -1 is [0 0 0 1]\'

Two linearly independent vectors for 2 are [0 1 2 0]\' and [1 1 2 0]\'

(Here \' stands for transpose).

Writing down these column vectors we obtain P and

P^-1 AP = diag (1,-1,2,2)

So A¹⁰ = P diag (1,1,1024, 1024) P^-1 , which can be easily evaluated.