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My question is similar to this one:

Orthogonal complement of the diagonal matrices in the inner product space of matrices

Considering $\mathbb{R}^{n\times n}$ with inner product $\langle A,B \rangle = \text{tr} (AB)$. Describe the orthogonal complement of the subspace of $\mathbb{R}^{n\times n}$ with all the diagonal matrices.

So my thoughts are similar to the ones of @LiúJiāgěng. The orthogonal complement of the subspace $U \le V$ consisting of the vectors of the space $V$ that verify $\langle u,v\rangle=0$ for all $v$. Therefore, the only matrices that verify this for all the diagonal matrices are the matrices with only zeros in the diagonal.

Can you please verify if I'm right? I don't know if that user's answer is correct or not, because nothing was said about it, nor was it upvoted. Thanks!

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WatsonHolmes
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    This is the correct answer, but surely something more should be said here to justify "therefore". One way to do this is to pick a basis $(E_a)$ of the diagonal matrices and see what conditions $\langle E_a, A \rangle = 0$ imposes on a general matrix $A$. – Travis Willse May 26 '16 at 14:56

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