ISBN-10:
0821846108
ISBN-13:
9780821846100
Pub. Date:
07/14/2008
Publisher:
American Mathematical Society
Lectures on Matrices

Lectures on Matrices

by J. H. M. Wedderburn
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Product Details

ISBN-13: 9780821846100
Publisher: American Mathematical Society
Publication date: 07/14/2008
Series: Colloquium Publications Series , #17
Edition description: Reprint
Pages: 205
Product dimensions: 7.00(w) x 10.00(h) x 0.50(d)

Table of Contents

Prefaceiii
Chapter IMatrices and Vectors
Section 1Linear transformations and vectors1
Section 2Linear dependence2
Section 3Linear vector functions and matrices3
Section 4Scalar matrices5
Section 5Powers of a matrix; adjoint matrices6
Section 6The transverse of a matrix8
Section 7Bilinear forms9
Section 8Change of basis9
Section 9Reciprocal and orthogonal bases11
Section 10The rank of a matrix14
Section 11Linear dependence16
Chapter IIAlgebraic Operations with Matrices. The Characteristic Equation
Section 1Identities20
Section 2Matric polynomials in a scalar variable20
Section 3-4The division transformation21
Section 5-6The characteristic equation23
Section 7-8Matrices with distinct roots25
Section 9-12Matrices with multiple roots27
Section 13The square root of a matrix30
Section 14Reducible matrices31
Chapter IIIInvariant Factors and Elementary Divisors
Section 1Elementary transformations33
Section 2The normal form of a matrix34
Section 3Determinantal and invariant factors36
Section 4Non-singular linear polynomials37
Section 5Elementary divisors38
Section 6-7Matrices with given elementary divisors39
Section 8-9Invariant vectors43
Chapter IVVector Polynomials. Singular Matric Polynomials
Section 1Vector polynomials47
Section 2The degree invariants48
Section 3-4Elementary sets49
Section 5Linear elementary bases52
Section 6Singular linear polynomials55
Chapter VCompound Matrices
Section 1Compound matrices63
Section 2The scalar product63
Section 3Compound matrices64
Section 4Roots of compound matrices67
Section 5Bordered determinants67
Section 6-7The reduction of bilinear forms68
Section 8Invariant factors71
Section 9Vector products72
Section 10The direct product74
Section 11Induced or power matrices75
Section 12-14Associated matrices76
Section 15Transformable systems79
Section 16-17Transformable linear sets80
Section 18-19Irreducible transformable sets85
Chapter VISymmetric, Skew, and Hermitian Matrices
Section 1Hermitian matrices88
Section 2The invariant vectors of a hermitian matrix90
Section 3Unitary and orthogonal matrices91
Section 4Hermitian and quasi-hermitian forms92
Section 5Reduction of a quasi-hermitian form to the sum of squares93
Section 6The Kronecker method of reduction96
Section 7Cogredient transformation98
Section 8Real representation of a hermitian matrix100
Chapter VIICommutative Matrices
Section 1Commutative matrices102
Section 2Commutative sets of matrices105
Section 3Rational methods106
Section 4The direct product108
Section 5Functions of commutative matrices110
Section 6Sylvester's identities111
Section 7Similar matrices113
Chapter VIIIFunctions of Matrices
Section 1Matric polynomials115
Section 2Infinite series115
Section 3The canonical form of a function116
Section 4Roots of 0 and 1118
Section 5-6The equation y[superscript m] = x; algebraic functions119
Section 7The exponential and logarithmic functions122
Section 8The canonical form of a matrix in a given field123
Section 9The absolute value of a matrix125
Section 10Infinite products127
Section 11The absolute value of a tensor127
Section 12Matric functions of a scalar variable128
Section 13Functions of a variable vector130
Section 14Functions of a variable matrix135
Section 15-16Differentiation formulae136
Chapter IXThe Automorphic Transformation of a Bilinear Form
Section 1Automorphic transformation140
Section 2-3The equation y' = [plus or minus]aya[superscript -1]141
Section 4Principal idempotent and nilpotent elements142
Section 5The exponential solution144
Section 6Matrices which admit a given transformation145
Chapter XLinear Associative Algebras
Section 1Fields and algebras147
Section 2Algebras which have a finite basis148
Section 3The matric representation of an algebra149
Section 4The claculus of complexes150
Section 5The direct sum and product151
Section 6Invariant subalgebras152
Section 7Idempotent elements154
Section 8-9Matric subalgebras156
Section 10-12The classification of algebras158
Section 13Semi-invariant subalgebras163
Section 14The representation of a semi-simple algebra165
Section 15Group algebras167
Appendix I
Notes169
Appendix II
Bibliography172
Index to bibliography194
Index197

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