Consider a set of operators $O$ on a Hilbert space $V$ of dimension $d$. I could prove that $O$ is also a Hilbert space with dimension $d^2$ (inner product being $(A,B) = tr(A^\dagger B))$. Now I am trying to find an orthonormal basis of Hermitian matrices for the space of $O$. However I am not sure how to proceed, also cant understand how Gram-Schmidt can help here.
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you know that $V$ is isomorphic to $\mathbb C^d$? – user251257 Feb 20 '16 at 20:26
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@user251257 I dont understand how that helps. – Anuroop Kuppam Feb 21 '16 at 03:56
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Do you know a basis for hermitian matrices? – user251257 Feb 21 '16 at 04:31
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@user251257 Nope! I am not sure we are on the same page. I am trying to find an orthonormal basis for $O$, which constitutes of only Hermitian matrices. – Anuroop Kuppam Feb 21 '16 at 04:45
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Once you have a basis you can obtain an orthonormal one. Btw. The set of complex hermitian matrices is not a complex vector space. However it is a real one. – user251257 Feb 21 '16 at 05:19