Question 1168387
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If A = ({{{a[i,j]}}})  and  B = ({{{b[i,j]}}}),  then 


    (A,B) = tr(A^t * B) = {{{sum (a[i,j]*b[i,j], 1,n)}}}.



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 = ({{{a[i,j]}}}) | {{{sum (a[i,i], 1, n) = 0}}} }.     <U>ANSWER</U>



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