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Are there terminologies for $A A^T$ and $A^T A$, respectively, where $A$ is a matrix?

Like "$A A^T$ is the (something) of $A$."

I know that if $A$ were a vector, we could use the terms inner product and outer product, but they seem not appropriate for matrices. Also, if we use them, they are the inner/outer product of $A$ and $A$, so that we should use the $A$ twice.

Likewise are there terminologies for $A A^H$ and $A^H A$, respectively, where $A$ is a complex matrix?

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I don't know of names for those expressions in general, however they both show up in particular contexts. (I'll assume all real numbers)

If a sequence of vectors are arranged as the columns of $A$, then $A^{T}A$ is the Gramian of the sequence of vectors.

If a sequence of multivariate observations are arranged as the columns of $A$, then $AA^{T}$ is the uncentered and unscaled sample covariance of the data.

Note that in both contexts, different authors may arrange vectors as rows, or observations as rows, thereby turning $AA^{T}$ into $A^{T}A$ and vice versa.

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