A Survey of Matrix Theory and Matrix Inequalities by Marvin Marcus Henryk Minc

Author:Marvin Marcus, Henryk Minc
Language: eng
Format: epub
Publisher: Dover Publications

and while , a contradiction.

5.7.3 If A and B are symmetric matrices in Mn(R), then A and B are congruent over R, if and only if they have the same rank and index.

5.7.4 If A ∈ Mn(K) and AT = – A (A is skew-symmetric), then ρ(A) = 2p and A is congruent over K to a matrix of the form

in which q1, q1, q2, q2, ···, qp, qp are the invariant factors of A. Thus two skew-symmetric matrices A and B are congruent over K, if and only if they are equivalent (see 3.17.3) over K.

5.8 Hermitian congruence

In case K = C the related notion of hermitian congruence arises. If A and B are in Mn(C), then A and B are hermitely congruent or conjunctive, if there exists a nonsingular P ∈ Mn(C) such that A = P*BP. Note that matrices conjunctive to hermitian (skew-hermitian) matrices are hermitian (skew-hermitian).

5.8.1 Suppose A and B are conjunctive over C, A = P*BP. Then P is a product of elementary matrices, P = P1 ··· Pm so that

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