## Transpose of a Matrix and Eigenvalues and Related Questions

## Problem 12

Let $A$ be an $n \times n$ real matrix. Prove the followings.

**(a)** The matrix $AA^{\trans}$ is a symmetric matrix.

**(b) **The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal.

**(c)** The matrix $AA^{\trans}$ is non-negative definite.

(An $n\times n$ matrix $B$ is called *non-negative definite* if for any $n$ dimensional vector $\mathbf{x}$, we have $\mathbf{x}^{\trans}B \mathbf{x} \geq 0$.)

**(d)** All the eigenvalues of $AA^{\trans}$ is non-negative.