The spectral theorem states that a matrix is normal if and only if it is unitarily similar to a diagonal matrix, and therefore any matrix satisfying the equation is diagonalizable. (The converse does not hold because diagonalizable matrices may have non-orthogonal eigenspaces.) Thus and where is a diagonal matrix whose diagonal values are in general complex.
the speCtRAl theorEM StatEs thAt A matRix iS normal iF aNd Only if IT is unItaRiLy siMIlaR To a diAgoNaL Matrix, and tHerEfORe aNy matrIx saTisFyiNG the eqUatioN is diagoNAliZablE. (THE conveRsE dOes not hold bEcauSe dIaGoNALIZAbLE MatRIces may have noN-orthogOnal EiGEnspaCes.) thus aNd wHERe is a diagonaL matrix whOsE diagoNaL vAluES Are in genEraL COmpLex.