I know that $\Tr(X^H Y)$ is a inner product, what about $\Tr(X^H YP)$? I found a physics paper (https://arxiv.org/pdf/2403.04199) that claims it is given that P is positive definite, but cannot find any proof
#Is <X,Y>_P := \Tr(X^H YP) is an inner product?
sharp crow
prisma sailBOT
native warrenBOT
Juan Carlos
sharp crow
Nevermind, for positive definiteness we can Cholesky factorize it as $P=L^H L$, then
[\Tr(X^H XL^H L)=\Tr(LX^H XL^H)=\Tr((XL^H)^H(XL^H))]
Is that legit
Then uhhhhhh
native warrenBOT
Juan Carlos
sharp crow
We have that $\Tr((XL^H)^H(XL^H))=||{XL^H}||^2$, under Frobenius norm
Since that is a norm (not a seminorm), if that is equal to 0 then $XL^H=0$, and then $X=0$ since $L$ is invertible
oak narwhal
obtuse creek
an inner product is defined over a vector space of matrices, so for addition to make sense they should be the same size
inner products must be conjugate symmetric, so both X^H Y and Y^H X should be defined (i.e. X and Y are square)
sharp crow
an inner product is defined over a vector space of matrices, so for addition to ...
X and Y are the same dimension so that's fine