I know that $x^TAy = y^TAx$ is true for symmetric quadratic matrices, but, it is true for non symmetric quadratic matrices?
2026-04-04 08:42:39.1775292159
Is $x^TAy = y^TAx$ for any matrix $A$?
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No. Let $e_i$ denotes the vector with a $1$ at the $i$-th position and zeros elsewhere. If $A$ is not symmetric, then $a_{ij}\ne a_{ji}$ for some $i\ne j$, but then $e_i^TAe_j=a_{ij}\ne a_{ji}=e_j^TAe_i$.