SOLUTION: Let M22 have the inner product ⟨A, B⟩ = tr(A^T B). Describe the orthogonal complement of the subspace of all DIAGONAL and SYMMETRIC matrices.

If A = ()  and  B = (),  then 


    (A,B) = tr(A^t * B) = .



From it, it is easy to deduce that the orthogonal complement of the subspace of all DIAGONAL matrices

is the subspace of the matrices with the zero trace     { A = () |  }.     ANSWER



Similarly, it is easy to deduce that the orthogonal complement of space of symmetric matrices is the space of all skew-symmetric matrices.