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algebra application bilinear form chain Chapter classical canonical form classical form coeﬁicients column vectors commutes complex conﬂuent congruent transformation consider convergence corresponding deﬁned degree denotes determinant diagonal elements diﬁerent diﬁerential eﬁect elementary divisors equal equations example ﬁeld ﬁnal ﬁnd ﬁrst column follows given Hence Hermitian form Hermitian matrix homogeneous identity integers invariant factors isolated latent point Lemma linearly independent Math matrix H matrix of order matrix pencil minimal numbers minor of order non-negative deﬁnite non-singular matrix non-zero elements notation null nullity obtain operation of Type orthogonal matrix pairs Paperbound partitioned polynomial positive deﬁnite principal minors problem proof properties prove quadratic form quadric rank rational canonical form rectangular rows and columns satisﬁed scalar function Segre characteristic singular pencil skew symmetric solution square matrix submatrices superdiagonal symmetric matrix theorem theory unit matrix unitary matrix values vanish variables veriﬁed zero