Let $A$ be a square complex matrix. What special properties are possessed by $AA^H$, where $^H$ denotes the conjugate transpose?
One property I am aware of is that $AA^H$ is Hermitian, i.e. $AA^H=(AA^H)^H$ – in fact, this is true even when $A$ is not square.
Are there any other special properties of $AA^H$?
Best Answer
The essential property is that $B=A^HA$ (I prefer this way, more natural) is "symmetrical semi-definite positive", with, as a consequence, all its eigenvalues real and $\geq 0$. If $A$ is full-rank, $B$ is definite positive (all its eigenvalues real and $>0$).
Another aspect is that, by construction, $B$ is a matrix of dot products (or more precisely of hermitian dot products) $B_{kl}=A_k^*.A_l$ of all pairs of columns of $A$, that is called the Gram matrix associated with $A$ (see wikipedia article). Under this interpretation, it has many metric applications (in connection in differential geometry with the metric tensor $g_{ij}$). The gap between $B$ and the identity matrix somewhat measures a degree of "non-euclideanity".