# What is the meaning of $A^TA$?

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.

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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.