document.write( "Question 1168387: 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.
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Algebra.Com's Answer #792978 by ikleyn(52794) $\"\"$ $\"About$

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document.write( "If A = ()  and  B = (),  then \r\n" );
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document.write( "    (A,B) = tr(A^t * B) = .\r\n" );
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document.write( "From it, it is easy to deduce that the orthogonal complement of the subspace of all DIAGONAL matrices\r\n" );
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document.write( "is the subspace of the matrices with the zero trace     { A = () |  }.     ANSWER\r\n" );
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document.write( "Similarly, it is easy to deduce that the orthogonal complement of space of symmetric matrices is the space of all skew-symmetric matrices.\r\n" );
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