## An Introduction to the Theory of Canonical MatricesThorough and self-contained, this penetrating study of the theory of canonical matrices presents a detailed consideration of all the theory’s principal features — from definitions and fundamental properties of matrices to the practical applications of their reduction to canonical forms. Beginning with matrix multiplication, reciprocals, and partitioned matrices, the text proceeds to elementary transformations and bilinear and quadratic forms. A discussion of the canonical reduction of equivalent matrices follows, including treatments of general linear transformations, equivalent matrices in a field, the H. C. F. process for polynomials, and Smith’s canonical form for equivalent matrices. Subsequent chapters treat subgroups of the group of equivalent transformations and collineatory groups, discussing both rational and classical canonical forms for the latter. Examinations of the quadratic and Hermitian forms of congruent and conjunctive transformative serve as preparation for the methods of canonical reduction explored in the final chapters. These methods include canonical reduction by unitary and orthogonal transformation, canonical reduction of pencils of matrices using invariant factors, the theory of commutants, and the application of canonical forms to the solution of linear matrix equations. The final chapter demonstrates the application of canonical reductions to the determination of the maxima and minima of a real function, solving the equations of the vibrations of a dynamical system, and evaluating integrals occurring in statistics. |

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An Introduction to the Theory of Canonical Matrices H. W. Turnbull,A. C. Aitken Limited preview - 2004 |

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algebra altemant arbitrary 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 edition elementary divisors equal equation example ﬁeld ﬁgures ﬁnal ﬁnd ﬁrst column follows given Hence Hermitian form Hermitian matrix identity integers invariant factors isolated latent point Lemma linear transformation linearly independent Math matrix H matrix of order matrix pencil minimal numbers minor of order multiplication non-negative deﬁnite non-singular matrix non-zero elements notation null obtain operation of Type orthogonal matrix pairs partitioned permutable polynomial positive deﬁnite principal minors properties prove quadratic form quadric rank rational canonical form reciprocal rectangular rows and columns satisﬁed Segre characteristic singular pencil skew symmetric solution square matrix submatrices superdiagonal symmetric matrix theorem theory transposed Unabridged republication unit matrix unitary matrix values vanish variables veriﬁed zero