Ejercicio Problem 2. Eigenvalues, eigenvectors and matrix p… · Matemàtiques URV Exemples 2024

Eigenvalues of A (smaller first, separated by comma)

Solución: 0, 2

a. Find the characteristic polynomial, the eigenvalues, and the corresponding eigenspaces of A.

Solución: p(lambda) = -lambda*(2-lambda)^2, eigenvalues lambda_1 = 2 (mult 2) and lambda_2 = 0 (mult 1); E_{lambda=2} = span{(1,0,1), (0,1,0)}, E_{lambda=0} = span{(-1,0,1)}

Value of the entry (A^n)_{1,1}

Solución: 2^(n-1)

b. Compute A^n for any positive integer n >= 1.

Solución: A^n = [[2^(n-1), 0, 2^(n-1)], [0, 2^n, 0], [2^(n-1), 0, 2^(n-1)]]