What are the eigenvalues and eigenvectors of $A^2 - 3A + 4I$, given the eigenvalues and eigenvectors of $A$?

Question

Let eigenvalues of $2 \times 2$ matrix $A$ be $1,-2$ and eigenvectors be $x_1$ & $x_2$ respectively. Then eigenvalues and eigenvectors of $A^2-3A+4I$ would be?

We know that eigenvalues can be calculated by substituting in the equation of new matrix. But what is the relation of eigenvectors with new matrix in such cases?

Answer 1

Hint: if $A v = \lambda v$, and $P$ is a polynomial, then $P(A) v = P(\lambda) v$.

Answer 2

The eigenvectors stay the same. For convenience take the eigenvalue 1 and eigenvector x1. Then, ((A^2) - 3A + 4I) .x1=((1) ^2)x1 - 3.(1.x1)+ 4 (x1) =2.x1. Similarly the others.