Is there an intuitive explaination of the matrix $A^T A$? I have seen this in many field and it is also a matrix with a lot of good properties. Is there some intuitive explaination of it or a name for it?
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These matrices are sometimes referred to as Gram matrices.
One way to think about it is to note that the $i,j$ entry of $A^TA$ is the dot-product of columns $i$ and $j$ from $A$.
Most of the "nice properties" come out of the fact that $A^TA$ is always positive semidefinite (and symmetric). Conversely, every (symmetric) positive semidefinite matrix is a Gram matrix.
Ben Grossmann
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"Every positive semidefinite matrix is a Gram matrix" holds if you are defining positive semidefinitness over $\mathbb C^n$. If you define it over $\mathbb R^n$, it is not true. – Martin Argerami Oct 22 '14 at 04:21
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