Name of the matrix transform $AA^*$ given A?

Question

There are a number of places this matrix transform making its appearance:

1. Every positive semi-definite matrix $B$ can have a decomposition $B=AA^*$

2. If the matrix $A$ is a lower triangular matrix then $B=AA^*$ is also called as Cholesky decomposition of $B$

3. Given a system of linear equations: $Ax=b$, $(A^*A)x = (A^*b)$ represents "normal equations"

4. If X is a random vector, $\mathbb{E}[XX^*]$ is called its covariance matrix

5. Poalr decomposition problem : A = UP (Unitary and Positive semi-definite) is solved by $P=\sqrt{A^*A}$ (Courtesy: Jonas Meyer)

... and so on.

I suspect there is a name for the tranformation $A^*A$ or $AA^*$!

Is there a name really? What would you suggest if there is n't?

A matrix of normal equations ?? (Doesn't look nice!)

Answer 1

The Gramian or Gram Matrix looks like what you're asking for. As JonasMeyer points out, you have to take into consideration whether you want the Gram matrix of the row or the column vectors.

(Subjective addendum: To simplify terminology, instead of saying "the Gramian of the columns of $A$" and "the Gramian of the rows of $A$," you could call $A^TA$ and $AA^T$ something like the column-Gramian and the row-Gramian, respectively. You could take the simplification one step further by calling one of these the Gramian, and only requiring the prefix for the other one (similar to what we do with eigenvectors: there are right and left eigenvectors, but "eigenvector" on its own usually means right eigenvector.))