This is a bilinear form on nxn matrices. Is it positive definite?
2026-03-31 21:12:06.1774991526
How do you prove that (A,B) = trace(AB) is an inner product?
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It's certainly not the case that $Tr(A^2)\ge 0$ for all $A$. For example, if $A$ is the 2x2 matrix giving a 90 degree rotation of the plane, then $Tr(A^2)$ is $-2$. Perhaps you want to define $(A,B)=Tr(A^{T}B)$ instead? That makes $(A,A)\ge 0$ easy to show (for $A$ real) since it's just the sum of the squares of the entries of $A$. For $A$ complex, use the conjugate transpose instead of the ordinary transpose, and you will be fine then, too.