What does it mean if a matrix is multiplied by its transpose? Informally, it seems like $A^TA$ boils a matrix down to its essentials, but can this operation somehow be understood "intuitively" (e.g. through a geometric interpretation)?
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To elaborate on user9325's answer, the standard inner product of two vectors $x, y$ is given by $x^T y$. If we change coordinates $x \mapsto Ax', y \mapsto Ay'$, then the inner product becomes $x^T y \mapsto x'^T (A^T A) y'$, so the matrix $A^T A$ encodes the coefficients of the new inner product.
I will assume that your matrix is real.
1) If $A$ encodes a change of basis, you can ask what happens to the standard inner product of two vectors. The matrix above gives you the formula for this product in the new basis.
2) The eigenvalues of your product are the singular values of the matrix (in particular, it does not have to be a square matrix for this interpretation). Note that the product "makes the matrix square and symmetric".
(which also explains that this gives the length of the semi-axes of an ellipse in a special case)