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.
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