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Suppose we have an inner product defined by $\langle x,y\rangle = x^TAy $. Is there any intuition for describing this inner product? Might it correspond to the dot product with a different basis? Is it related to the transformation described by the matrix, and if so, how? Thanks.

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Note that the map $\langle x,y \rangle = x^TAy$ is an inner product if and only if $A$ is symmetric and positive definite. Notably, $A$ is symmetric and positive definite if and only if there exists an invertible matrix $M$ such that $A = M^TM$.

With that in mind: if we suppose that your bilinear map is an inner product, then let $M$ be an invertible matrix such that $A = M^TM$. We note that $$ x^TAy = x^TM^TMy = (Mx)^T(My) $$ To put it another way: we apply the change of basis $M$ to the vectors $x$ and $y$, then take the dot product of the result. You may find it a useful exercise to verify that the columns of $M^{-1}$ will form an orthonormal basis with respect to this inner product.

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