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

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?

• Try computing $(A^2-3A+4I)x_1$ and $(A^2-3A+4I)x_2$, and see what that tells you. – JimmyK4542 Sep 23 '19 at 3:30

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