# Positive-semi-definite matrix

1. Nov 19, 2008

### Cyrus

$$A*A^T$$

Will result in a symmetric matrix. Does this mean that A*A^T will be positive semi-definite?

Does A have to be strictly real, can it be complex?

2. Nov 20, 2008

### D H

Staff Emeritus
A*AT is symmetric IF A is real. For complex matrices you want to use the adjoint rather than the transpose.

The eigenvalues of A*A* are always real and non-negative. The square roots of these eigenvalues are called the singular values of A. Singular value decomposition is a very important matrix decomposition technique.