SCHAUM'S THEORY aiMl>ROBLEMS OUTLINE SERIES of MATRICES m o c . by FRANK AYRES, JR. l a u n a M n o i t u l o S e including h T . w w w Completely Solved in Detail SCHAUM PUBLISHING CO. NEW YORK SCHAVM'S OUTLINE OF THEORY PROBLEMi§; AI\[D OF MATRICES m o c . l a u n a M n o i t u BY l o S FRANK AYRES, JR., Ph.D. e h T Formerly Professorand Head, . Departmentof Mathematics w w Dickinson College w ^C^O -^ SCHAIJM'S OUTLINE SERIES McGRAW-HILL BOOK COMPANY New York, St. Louis, SanFrancisco, Toronto, Sydney m o ('opyright © 1962 by McGraw-Hill, Inc. AH Rights Reserved. Printed in the c . United States of America. No part of this publication may be reproduced, l stored in a retrieval system, or transmitted, in any form or by any means, a u electronic, mechanical, photocopying, recording, or otherwise, without the prior n written permission of the publisher. a M 02656 n 78910 SHSH 7543210 o i t u l o S e h T . w w w Preface Elementary matrix algebra has now become an integral partofthe mathematical background necessary for such diverse fields as electrical engineering and education, chemistry and sociology, as well as for statistics and pure mathematics. This book, in presenting the more essential mate- rial, is designed primarily to serve as a useful supplementto current texts and as ahandy refer- ence book for those working in the several fields which require some knowledge ofmatrix theory. Moreover, the statements of theory and principle are sufficiently complete that the book could be used as a text by itself. The material has been divided into twenty-six chapters, since the logical arrangement is om thereby not disturbed while the usefulness as a reference book is increased. This also permits c a separation of the treatment of real matrices, with which the majority of readers will be con- . cerned, from that of matrices with complex elements. Each chapter contains a statement of perti- l nent definitions, principles, and theorems, fully illustrated by examples. These, in turn, are ua followed by a carefully selected set of solved problems and a considerable number of supple- n mentary exercises. a M The beginning student in matrix algebra soon finds that the solutions of numerical exercises on are disarmingly simple. Difficulties are likely to arise from the constant round of definition, the- i orem, proof. The trouble here is essentially a matter of lack of mathematical maturity,' and t u normally to be expected, since usually the student's previous work in mathematics has been l concerned with the solution of numerical problems while precise statements of principles and o proofs of theorems have in large part been deferred for later courses. The aim of the present S book is to enable the reader, if he persists through the introductory paragraphs and solved prob- he lems in any chapter, to develop a reasonable degree of self-assurance about the material. T . The solved problems, in addition to giving more variety to the examples illustrating the w w theorems, contain most of the proofs of any considerable length together with representative w shorter proofs. The supplementary problems call both for the solution of numerical exercises and for proofs. Some of the latter require only proper modifications of proofs given earlier; more important, however, are the many theorems whose proofs require but a few lines. Some are of the type frequently misnamed "obvious" while others will be found to call for considerable ingenuity. None should be treated lightly, however, for it is due precisely to the abundance of such theorems that elementary matrix algebra becomes a natural first course for those seeking to attain a degree of mathematical maturity. While the large number of these problems in any chapter makes it impractical to solve all of them before moving to the next, special attention is directed to the supplementary problems ofthe first two chapters. A mastery of these will do much to give the reader confidence to stand on his own feet thereafter. The author wishes to take this opportunity to expresshis gratitude to the staffof the Schaum Publishing Company for their splendid cooperation. Frank Ayres, Jr. CarHsle, Pa. October, 1962 m o c . l a u n a M n o i t u l o S e h T . w w w CONTENTS Page Chapter 1 MATRICES 1 Matrices. Equal matrices. Sums of matrices. Products of matrices. Products by partitioning. 2 SOME TYPES OF MATRICES Chapter 10 Triangular matrices. Scalar matrices. Diagonal matrices. The identity m matrix. Inverse of a matrix. Transpose of a matrix. Symmetric o matrices. Skew-symmetric matrices. Conjugate of a matrix. Hermitian c matrices. Skew-Hermitian matrices. Direct sums. . l a u Chapter 3 DETERMINANT OF A SQUARE MATRIX. 20 n a Determinants of orders 2 and 3. Properties of determinants, Minors M and cofactors. Algebraic complements. n o i t Chapter 4 EVALUATION OF DETERMINANTS 32 u l Expansion along a row or column. The Laplace expansion. Expansion o along the first row and column. Determinant of a product. Derivative S of a determinant. e h T . Chapter 5 EQUIVALENCE 39 w w Rank of a matrix. Non-singular and singular matrices. Elementary w transformations. Inverse of an elementary transformation. Equivalent matrices. Row canonical form. Normal form. Elementary matrices. Canonical sets under equivalence. Rank of a product. Chapter 6 THE ADJOINT OF A SQUARE MATRIX 49 The adjoint. The adjoint of a product. Minor of an adjoint. Chapter 7 THE INVERSE OF A MATRIX. 55 Inverse of a diagonal matrix. Inverse from the adjoint. Inverse from elementary matrices. Inverse by partitioning. Inverse of symmetric m matrices. Right and left inverses of X w matrices. Chapter 8 FIELDS 64 Number fields. General fields. Sub-fields. Matrices over a field. CONTENTS Page Chapter 9 LINEAR DEPENDENCE OF VECTORS AND FORMS 67 Vectors. Linear dependence of vectors, linear forms, polynomials, and matrices. Chapter 10 LINEAR EQUATIONS 75 System of non-homogeneous equations. Solution using matrices. Cramer's rule. Systems of homogeneous equations. Chapter 11 VECTOR SPACES 85 Vector spaces. Sub-spaces. Basis and dimension. Sum space. Inter- section space. Null space of a matrix. Sylvester's laws of nullity. Bases and coordinates. m o c Chapter 12 LINEAR TRANSFORMATIONS 94 . l Singular and non-singular transformations. Change of basis. Invariant a space. Permutation matrix. u n a M Chapter 13 VECTORS OVER THE REAL FIELD 100 n o Inner product. Length. Schwarz inequality. Triangle inequality. i Orthogonal vectors and spaces. Orthonormal basis. Gram-Schmidt t orthogonalization process. The Gramian. Orthogonal matrices. Orthog- u onal transformations. Vector product. l o S e Chapter 14 VECTORS OVER THE COMPLEX FIELD 110 h T Complex numbers. Inner product. Length. Schwarz inequality. Tri- . angle inequality. Orthogonal vectors and spaces. Orthonormal basis. w Gram-Schmidt orthogonalization process. The Gramian. Unitary mat- w rices. Unitary transformations. w Chapter 15 CONGRUENCE 115 Congruent matrices. Congruent symmetric matrices. Canonical forms of real symmetric, skew-symmetric, Hermitian, skew-Hermitian matrices under congruence. Chapter 16 BILINEAR FORMS 125 Matrix form. Transformations. Canonical forms. Cogredient trans- formations. Contragredient transformations. Factorable forms. Chapter 17 QUADRATIC FORMS 131 Matrix form. Transformations. Canonical forms. Lagrange reduction. Sylvester's law of inertia. Definite and semi-definite forms. Principal minors. Regular form. Kronecker's reduction. Factorable forms. CONTENTS Page Chapter lo HERMITIAN FORMS 146 Matrix form. Transformations. Canonical forms. Definite and semi- definite forms. Chapter lif THE CHARACTERISTIC EQUATION OF A MATRIX 149 Characteristic equation and roots. Invariant vectors and spaces. Chapter 20 SIMILARITY 156 Similar matrices. Reduction to triangular form. Diagonable matrices. Chapter 21 SIMILARITY TO A DIAGONAL MATRIX 163 m o Real symmetric matrices. Orthogonal similarity. Pairs of real quadratic c forms. Hermitian matrices. Unitary similarity. Normal matrices. . Spectral decomposition. Field of values. l a u n 22 POLYNOMIALS OVER A FIELD a Chapter 172 M Sum, product, quotient of polynomials. Remainder theorem. Greatest n common divisor. Least common multiple. Relatively prime polynomials. o Unique factorization. i t u l Chapter 2o LAMBDA MATRICES 179 So The X-matrix or matrix polynomial. Sums, products, and quotients. e Remainder theorem. Cayley-Hamilton theorem. Derivative of a matrix. h T . w 24 SMITH NORMAL FORM w Chapter 188 w Smith normal form. Invariant factors. Elementary divisors. 25 THE MINIMUM POLYNOMIAL OF A MATRIX Chapter 196 Similarity invariants. Minimum polynomial. Derogatory and non- derogatory matrices. Companion matrix. Chapter 26 CANONICAL FORMS UNDER SIMILARITY 203 Rational canonical form. A second canonical form. Hypercompanion matrix. Jacobson canonical form. Classical canonical form. A reduction to rational canonical form. INDEX 215 INDEX OF SYMBOLS 219 m o c . l a u n a M n o i t u l o S e h T . w w w