How to check if matrix is diagonalizable?

Question

Is there any exact ways to check if a matrix is diagonalizable? For instance, I have an $m\times n$ matrix $A$ and I need to show that $AA^T$ is diagonalizable.

Answer 1

$AA^T$ is a symmetric matrix so it is diagonalizable.

Answer 2

$AA^T$ is symmetrical, hence it is diagonalizable.

For a general square matrix, compute the algebraic multiplicity and the geometric multiplicity for each eigenvalue. They have to be equal for it to be diagonalizable.

Edit: thanks to Robert Israel's comment, I am assuming $A$ is real.

Answer 3

Assuming $A$ is a real matrix, note that

$(AA^T)^T = (A^T)^T A^T = AA^T; \tag{1}$

thus $AA^T$ is symmetric. But symmetric matrices are always diagonalizable . . .