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!