## Conditions ordering the characteristic roots of matrices |

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5X has property acteristic roots alge algebraically closed field ApAg-AgAp bA2m benius Property characteristic column vectors characteristic function characteristic row vector Clearly column space column vector associated commutative semi-simple algebra desired result det(A diagonal algebra diagonal element direct sum distinct a t division algebra over^T exists f(Ai finite set Frobenius Property gonal Hermite form homomorphism identity matrix independent characteristic row induction infinite number istic root Lemma linear combination linearly independent characteristic mapping matric algebra over^f McCoy condition nc linearly independent necessary and sufficient non-singular non-zero row vectors nullity number of distinct polynomial preceeding theorems preserves characteristic roots Prom Properties PQ radical of 51 relations 5»1 roots of Aj_ roots of f(A1 row space row vector associated satisfies conditions set of matrices shown Suppose Theorem 2.5 Theorem 3»2 total matric algebra vectors of H Wedderburn's Principal Theorem Williamson zero divisors Ъе