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Consider you have a rectangular matrix $X$ whose columns are vectors with some meaning, i.e. sampled signals.

If you evaluate the product $G=X^H X$ you obtain the Gramian matrix $G$.

If you evaluate the product $M=X X^H$, what is the name of the $M$ matrix? Does it have a particular name?

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The entries of the Gramian matrix are the inner products of the columns of $X$, whereas the entries of $M$ are the inner products of the rows of $X$. Equivalently, $M$ is the Gramian matrix of $X^H$.

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