Find A^999(4 2 3)T, if A is a 3 × 3 matrix and it has the eigenpairs (λ1, v1) = (1, (1 -1 0)T) and (λ2, v2) = (-1, (2 0 1)T)

Question

I thought about finding matrix A first using A = A=PDP^-1, but I only have 2 eigenpairs. Any idea?

Answer 1

Express vector (4,2,3)T as a linear combination of the 2 eigenvectors and we get (4,2,3)T = -2(v1) + 3(v2).

And then we use the fact that A^999(v) = λ^999(v) to find A^999(4,2,3)T, as the vector (4,2,3)T is a linear combination of the 2 eigenvectors.

A^999(4,2,3)T = A^999(-2(v1)) + A^999(3(v2))

              = (λ1)^999(-2(v1)) + (λ2)^999(3(v2))

              = (1)^999(-2(1 -1 0)T) + (-1)^999(3(2 0 1)T)

              = (-8 2 -3)T