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block diagonal Brenner canonical matrix canonical set characteristic vector space column vectors columns form complex field Corollary 1-3 corresponding decomposable diag diagonal blocks diagonal elements diagonal matrix diagonal unitary group direct sum distinct characteristic roots easily seen elements of G equation exists a unitary form a normal full unitary group gonal group G hence Hermitian matrices holds identity matrix implies induction hypothesis integer invariant under G Lemma Moreover n x k n x n matrices n-rowed square matrix non-derogatory non-singular matrix normal matrices normal unitary basis null matrix null spaces obtain ordered k-tuplet ordered set ordering relation orthogonal pairs permutation matrix polar form polynomial in normal positive definite Hermitian properties R-ordered rational canonical form representation F scalar set of matrices Specht sub-group of G sub-matrices Theorem 1-1 triangular matrix UctAU uniquely defined uniquely determined unitarily similar unitary matrix unitary transform