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A-matrix algebra alternant arbitrary bilinear form chain Chapter classical canonical form classical form coefficients column vectors commutes complex congruent transformation conjugate conjunctive transformation consider convergence corresponding deﬁnite denotes derived determinant diagonal elements elementary divisors equal equivalent transformations example ﬁeld ﬁrst follows given H'AH Hence Hermitian form Hermitian matrix homogeneous identity integers interchange invariant factors isolated latent point Lemma linear transformation linearly independent Math matrix H matrix of order matrix pencil minor of order multiplication non-singular matrix non-zero elements notation null obtain operation of Type orthogonal matrix pairs Paperbound partitioned permutable polynomial positive definite premultiplying proof properties prove quadratic form quadric rank rational canonical form reciprocal rectangular Reduced Characteristic Function relation rows and columns Segre characteristic singular pencil skew symmetric solution square matrix subgroup submatrices superdiagonal symmetric matrix theorem theory trans transposed unit matrix unitary matrix values vanish variables zero