Inverse of a positive semi-definite matrix?

[mikeph]

Hi,
If A is some nonsquare matrix that is possible rank-deficient, then am I right to understand that (A^T)(A) is a positive semidefinite matrix? Does there exist an inverse (A^T A)^-1?

Thanks for any help

 

[D H]

Any real nxm matrix A will have A^TA (and AA^T) be positive semidefinite.

Now let A be some matrix all of whose elements are zero. Obviously both A^TA and AA^T will also be zero matrices (but now square), and obviously, no inverse.

There's a world of difference between positive definite and positive semidefinite.