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How would you generalize the coordinates of an m x n matrix in an orthonormal basis of m x n matrices of the subspace of all complex m x n matrices? I think this is fairly straight forward to think about using vectors but I'm having trouble thinking about this in the context or an orthonormal matrix basis.

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The "standard" inner product on complex $m \times n$ matrices is the Hilbert-Schmidt inner product, $(A,B) = \text{trace}(A^H B)$ (H = Hermitian adjoint = conjugate transpose): reverse the order if you prefer your inner product to be conjugate-linear on the right. If $E_{ij}$ is the matrix whose $(i,j)$ entry is $1$ and all others are $0$, note that $(E_{ij},E_{kl}) = 0$ unless $i=k$ and $j=l$, so these form an orthonormal basis. The coordinates of a matrix in this basis are the matrix entries ($a_{ij}$ is the coordinate for basis member $E_{ij}$).

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