let ''a'' operator finite dimensional inner product space ''a'' said normal ''a'' sup sup ''a'' ''a a'' sup sup one show ''a'' normal unitarily diagonalizable schur decomposition ''a'' ''u u'' sup sup ''u'' unitary ''t'' upper triangular since ''a'' normal ''t t'' sup sup ''t'' sup sup ''t'' therefore ''t'' must diagonal since normal upper triangular matrices diagonal converse obvious words ''a'' normal exists unitary matrix ''u''
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