# Are there terminologies for ($A A^T$ or $A A^H$) and ($A^T A$ or $A^H A$)?

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?

• I think one of them could be referred to as the self adjoint operator of A. Oct 17, 2018 at 5:30
• Oct 17, 2018 at 5:46

## 1 Answer

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.