Exercise 7

If $A$ is a square matrix, remember that the characteristic polynomial of $A$ is $\det(A-\lambda I)$ and that the roots of the characteristic polynomial are the eigenvalues of $A$.

1. Explain why $A$ and $A^T$ have the same characteristic polynomial.

2. Explain why $A$ and $A^T$ have the same set of eigenvalues.

3. Suppose that $A$ is diagonalizable with diagonalization $A=PDP^{-1}$. Explain why $A^T$ is diagonalizable and find a diagonalization.