An Introduction to Linear AlgebraCourier Corporation, 3 déc. 2012 - 464 pages "The straight-forward clarity of the writing is admirable." — American Mathematical Monthly. This work provides an elementary and easily readable account of linear algebra, in which the exposition is sufficiently simple to make it equally useful to readers whose principal interests lie in the fields of physics or technology. The account is self-contained, and the reader is not assumed to have any previous knowledge of linear algebra. Although its accessibility makes it suitable for non-mathematicians, Professor Mirsky's book is nevertheless a systematic and rigorous development of the subject. Part I deals with determinants, vector spaces, matrices, linear equations, and the representation of linear operators by matrices. Part II begins with the introduction of the characteristic equation and goes on to discuss unitary matrices, linear groups, functions of matrices, and diagonal and triangular canonical forms. Part II is concerned with quadratic forms and related concepts. Applications to geometry are stressed throughout; and such topics as rotation, reduction of quadrics to principal axes, and classification of quadrics are treated in some detail. An account of most of the elementary inequalities arising in the theory of matrices is also included. Among the most valuable features of the book are the numerous examples and problems at the end of each chapter, carefully selected to clarify points made in the text. |
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Page vi
... functions of a square matrix Partitioned matrices LINEAR OPERATORS 4.1. 4.2. 4.3. 4.4. Change of basis in a linear 'manifold Linear operators and their representations Isomorphisms and automorphisms of linear manifolds Further instances ...
... functions of a square matrix Partitioned matrices LINEAR OPERATORS 4.1. 4.2. 4.3. 4.4. Change of basis in a linear 'manifold Linear operators and their representations Isomorphisms and automorphisms of linear manifolds Further instances ...
Page vii
... functions of matrices The minimum polynomial and the theorem of Cayley and Hamilton Estimates of characteristic roots Characteristic vectors 7.5. 7.6. VIII. ORTHOGONAL AND UNITARY MATRICES IX. 8.1. Orthogonal matrices 8.2. Unitary ...
... functions of matrices The minimum polynomial and the theorem of Cayley and Hamilton Estimates of characteristic roots Characteristic vectors 7.5. 7.6. VIII. ORTHOGONAL AND UNITARY MATRICES IX. 8.1. Orthogonal matrices 8.2. Unitary ...
Page viii
... functions The relation between matrix functions and matrix polynomials Systems of linear differential equations. PART. III. QUADRATIC. FORMS. XII. BILINEAR, QUADRATIC, AND HERMITIAN FORMS 306 312 316 327 330 341 343 PART I DETERMINANTS ...
... functions The relation between matrix functions and matrix polynomials Systems of linear differential equations. PART. III. QUADRATIC. FORMS. XII. BILINEAR, QUADRATIC, AND HERMITIAN FORMS 306 312 316 327 330 341 343 PART I DETERMINANTS ...
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... function sgna: (read: signum a:) is defined as 1 (x > 0) sgnz= 0 (x= 0) -1 (x < 0). Exnncrss: 1.1.1. Show that sgnx-sgny = 58“ £3/, and deduce that sgn:r1.sgna:,...sgn:v,c = sgn(:c1:v,...:ck). DEFINITION 1.1.3. (i) e()q,...,) ...
... function sgna: (read: signum a:) is defined as 1 (x > 0) sgnz= 0 (x= 0) -1 (x < 0). Exnncrss: 1.1.1. Show that sgnx-sgny = 58“ £3/, and deduce that sgn:r1.sgna:,...sgn:v,c = sgn(:c1:v,...:ck). DEFINITION 1.1.3. (i) e()q,...,) ...
Page 33
... estimates. An independent proof has been given by G. B. Price, Proc. Amer. Math. Soc. 2 (1951), 497-502. 19. Let a,_,(:c) (r,s = 1,...,n) be differentiable functions of. I,§1.6 TWO SPECIAL THEOREMS ON LINEAR EQUATIONS 33.
... estimates. An independent proof has been given by G. B. Price, Proc. Amer. Math. Soc. 2 (1951), 497-502. 19. Let a,_,(:c) (r,s = 1,...,n) be differentiable functions of. I,§1.6 TWO SPECIAL THEOREMS ON LINEAR EQUATIONS 33.
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algebra assertion assume automorphism basis bilinear form bilinear operator canonical forms characteristic polynomial characteristic roots characteristic vectors coefiicients commute complement complex numbers convergent coordinates Deduce defined DEFINITION denote determinant diagonal elements diagonal form diagonal matrix dimensionality E-operations edition equal EXERCISE field find finite fixed follows functions geometry Hence hermitian form hermitian matrix identity implies inequality infinite integers inverse isomorphic linear combination linear equations linear manifold linear transformation linearly independent matrix group matrix of order minimum polynomial multiplication non-singular linear transformation non-singular matrix numbers obtain orthogonal matrix permutation positive semi-definite possesses problems proof of Theorem prove quadratic form quadric rank real symmetric reduces representation represented respect result rotation satisfies scalar Show similar singular skew-symmetric matrix solution specified square matrix suppose symmetric matrix Theorem theory tions unique unit element unitary matrix values vanish variables vector space view of Theorem write zero