Elementary Matrix Algebra

Name: Elementary Matrix Algebra
Availability: OutOfStock
ISBN: 9780023559501

ISBN-10: 0023559500

ISBN-13: 9780023559501

Edition: 3rd 1973

Authors: Franz E. Hohn

List price: $34.50

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Description:

This treatment starts with basics and progresses to sweepout process for obtaining complete solution of any given system of linear equations and role of matrix algebra in presentation of useful geometric ideas, techniques, and terminology.

Book details

List price: $34.50
Edition: 3rd
Copyright year: 1973
Publisher: Prentice Hall PTR
Binding: Hardcover
Pages: 522
Language: English

Introduction to Matrix Algebra
Matrices	p. 1
Equality of Matrices	p. 2
Addition of Matrices	p. 3
Commutative and Associative Laws of Addition	p. 3
Subtraction of Matrices	p. 4
Scalar Multiples of Matrices	p. 6
The Multiplication of Matrices	p. 7
The Properties of Matrix Multiplication	p. 9
Exercises	p. 13
Linear Equations in Matrix Notation	p. 18
The Transpose of a Matrix	p. 20
Symmetric, Skew-Symmetric, and Hermitian Matrices	p. 22
Scalar Matrices	p. 24
The Identity Matrix	p. 26
The Inverse of a Matrix	p. 26
The Product of a Row Matrix into a Column Matrix	p. 29
Polynomial Functions of Matrices	p. 30
Exercises	p. 32
Partitioned Matrices	p. 41
Exercises	p. 46
Linear Equations
Linear Equations	p. 51
Three Examples	p. 52
Exercises	p. 57
Equivalent Systems of Equations	p. 60
The Echelon Form for Systems of Equations	p. 63
Synthetic Elimination	p. 66
Systems of Homogeneous Linear Equations	p. 70
Exercises	p. 73
Computation of the Inverse of a Matrix	p. 75
Matrix Inversion by Partitioning	p. 78
Exercises	p. 82
Number Fields	p. 83
Exercises	p. 85
The General Concept of a Field	p. 85
Exercises	p. 88
Vector Geometry in E[superscript 3]
Geometric Representation of Vectors in E[superscript 3]	p. 90
Operations on Vectors	p. 91
Isomorphism	p. 94
Length, Direction, and Sense	p. 94
Orthogonality of Two Vectors	p. 97
Exercises	p. 100
The Vector Equation of a Line	p. 102
The Vector Equation of a Plane	p. 105
Exercises	p. 109
Linear Combinations of Vectors in E[superscript 3]	p. 111
Linear Dependence of Vectors; Bases	p. 114
Exercises	p. 118
Vector Geometry in n-Dimensional Space
The Real n-Space R[superscript n]	p. 120
Vectors in R[superscript n]	p. 121
Lines and Planes in R[superscript n]	p. 122
Linear Dependence and Independence in R[superscript n]	p. 125
Vector Spaces in R[superscript n]	p. 126
Exercises	p. 127
Length and the Cauchy-Schwarz Inequality	p. 129
Angles and Orthogonality in E[superscript n]	p. 132
Half-Lines and Directed Distances	p. 134
Unitary n-Space	p. 135
Exercises	p. 138
Linear Inequalities	p. 141
Exercises	p. 147
Vector Spaces
The General Definition of a Vector Space	p. 149
Linear Combinations and Linear Dependence	p. 153
Exercises	p. 159
Basic Theorems on Linear Dependence	p. 164
Dimension and Basis	p. 167
Computation of the Dimension of a Vector Space	p. 172
Exercises	p. 174
Orthonormal Bases	p. 177
Exercises	p. 180
Intersection and Sum of Two Vector Spaces	p. 183
Exercises	p. 186
Isomorphic Vector Spaces	p. 187
Exercises	p. 190
The Rank of a Matrix
The Rank of a Matrix	p. 191
Basic Theorems About the Rank of a Matrix	p. 195
Matrix Representation of Elementary Transformations	p. 198
Exercises	p. 204
Homogeneous Systems of Linear Equations	p. 210
Nonhomogeneous Systems of Linear Equations	p. 217
Exercises	p. 221
Another Look at Nonhomogeneous Systems	p. 229
The Variables One Can Solve for	p. 231
Basic Solutions	p. 234
Exercises	p. 237
Determinants
The Definition of a Determinant	p. 239
Some Basic Theorems	p. 243
The Cofactor in det A of an Element of A	p. 247
Cofactors and the Computation of Determinants	p. 250
Exercises	p. 253
The Determinant of the Product of Two Matrices	p. 264
A Formula for A[superscript -1]	p. 270
Determinants and the Rank of a Matrix	p. 272
Solution of Systems of Equations by Using Determinants	p. 274
A Geometrical Application of Determinants	p. 278
Exercises	p. 280
More About the Rank of a Matrix	p. 288
Definitions	p. 292
The Laplace Expansion	p. 295
The Determinant of a Product of Two Square Matrices	p. 299
The Adjoint Matrix	p. 300
The Row-and-Column Expansion	p. 302
The Diagonal Expansion of the Determinant of a Matrix	p. 303
Exercises	p. 305
Linear Transformations
Mappings	p. 310
Linear Mappings	p. 313
Some Properties of Linear Operators on Vector Spaces	p. 317
Exercises	p. 320
Linear Transformations of Coordinates	p. 323
Transformation of a Linear Operator	p. 331
Exercises	p. 334
The Algebra of Linear Operators	p. 340
Groups of Operators	p. 342
Exercises	p. 345
Unitary and Orthogonal Matrices	p. 347
Exercises	p. 348
Unitary Transformations	p. 351
Orthogonal Transformations	p. 353
The Eulerian Angles	p. 354
The Triangularization of a Real Matrix	p. 356
Exercises	p. 360
Orthogonal Vector Spaces	p. 361
Exercises	p. 366
Projections	p. 367
Orthogonal Projections in U[superscript n]	p. 371
Exercises	p. 373
The Characteristic Value Problem
Definition of the Characteristic Value Problem	p. 375
Four Examples	p. 377
Two Basic Theorems	p. 381
Exercises	p. 383
The Characteristic Polynomial and Its Roots	p. 387
Similar Matrices	p. 392
Exercises	p. 393
The Characteristic Roots of a Hermitian Matrix	p. 395
The Diagonal Form of a Hermitian Matrix	p. 397
The Diagonalization of a Hermitian Matrix	p. 400
Examples	p. 401
Triangularization of an Arbitrary Matrix	p. 403
Normal Matrices	p. 404
Exercises	p. 406
Characteristic Roots of a Polynomial Function of a Matrix	p. 409
The Cayley-Hamilton Theorem	p. 411
The Minimum Polynomial of a Matrix	p. 413
Powers of Matrices	p. 418
Exercises	p. 419
Quadratic, Bilinear, and Hermitian Forms
Quadratic Forms	p. 422
Diagonalization of Quadratic Forms	p. 424
A Geometrical Application	p. 425
Definite Forms and Matrices	p. 428
Exercises	p. 433
Lagrange's Reduction	p. 438
Kronecker's Reduction	p. 443
Sylvester's Law of Inertia for Real Quadratic Forms	p. 445
A Necessary and Sufficient Condition for Positive Definiteness	p. 449
An Important Example	p. 452
Exercises	p. 454
Pairs of Quadratic Forms	p. 455
Values of Quadratic Forms	p. 457
Exercises	p. 462
Bilinear Forms	p. 463
The Equivalence of Bilinear Forms	p. 465
Cogredient and Contragredient Transformations	p. 467
Exercises	p. 469
Hermitian Forms	p. 471
Definite Hermitian Forms	p. 474
Exercises	p. 474
The Notations [Sigma] and [Pi]	p. 477
The Algebra of Complex Numbers	p. 491
Bibliography	p. 501
Index	p. 513
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