Matrix Theory: From Generalized Inverses to Jordan Form (Pure and Applied Mathematics: A Program of Monographs and Textbooks)

In 1990, the National Science Foundation recommended that every college mathematics curriculum should include a second course in linear algebra. In answer to this recommendation, Matrix Theory: From Generalized Inverses to Jordan Form provides the material for a second semester of linear algebra that probes introductory linear algebra concepts while also exploring topics not typically covered in a sophomore-level class. Tailoring the material to advanced undergraduate and beginning graduate students, the authors offer instructors flexibility in choosing topics from the book. The text first focuses on the central problem of linear algebra: solving systems of linear equations. It then discusses LU factorization, derives Sylvester's rank formula, introduces full-rank factorization, and describes generalized inverses. After discussions on norms, QR factorization, and orthogonality, the authors prove the important spectral theorem. They also highlight the primary decomposition theorem, Schur's triangularization theorem, singular value decomposition, and the Jordan canonical form theorem. The book concludes with a chapter on multilinear algebra. With this classroom-tested text students can delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra. Cover......Page 1 Title page......Page 6 Preface......Page 10 Introduction......Page 12 Contents......Page 16 1.1 Solving Systems of Linear Equations......Page 22 1.1.1.1 Floating Point Arithmetic......Page 31 1.1.1.2 Arithmetic Operations......Page 32 1.1.1.3 Loss of Significance......Page 33 1.1.2.1 Creating Matrices in MATLAB......Page 34 1.2 The Special Case of "Square" Systems......Page 38 1.2.1 The Henderson Searle Formulas......Page 42 1.2.2 Schur Complements and the Sherman-Morrison-Woodbury Formula......Page 45 1.2.3.1 Computing Inverse Matrices......Page 58 1.2.4.2 Operation Counts......Page 60 2.1 A Brief Review of Gauss Elimination with Back Substitution......Page 62 2.1.1.1 Solving Systems of Linear Equations......Page 68 2.2 Elementary Matrices......Page 70 2.2.1 The Minimal Polynomial......Page 78 2.3 The LU and LDU Factorization......Page 84 2.3.1.1 The LU Factorization......Page 96 2.4 The Ad jugate of a Matrix......Page 97 2.5 The Frame Algorithm and the Cayley-Hamilton Theorem......Page 102 2.5.1 Digression on Newton's Identities......Page 106 2.5.2 The Characteristic Polynomial and the Minimal Polynomial......Page 111 2.5.4.1 Polynomials in MATLAB......Page 116 5 Generalized Inverses I......Page 120 3.1.1.1 The Fundamental Subspaces......Page 130 3.2 A Deeper Look at Rank......Page 132 3.3 Direct Sums and Idempotents......Page 138 3.4 The Index of a Square Matrix......Page 149 3.4.1.1 The Standard Nilpotent Matrix......Page 168 3.5 Left and Right Inverses......Page 169 4.1 Row Reduced Echelon Form and Matrix Equivalence......Page 176 4.1.1 Matrix Equivalence......Page 181 4.1.2.1 Row Reduced Echelon Form......Page 188 4.1.3.1 Pivoting Strategies......Page 190 4.1.3.2 Operation Counts......Page 191 4.2 The Hermite Echelon Form......Page 192 4.3 Full Rank Factorization......Page 197 4.4 The Moore-Penrose Inverse......Page 200 4.5 Solving Systems of Linear Equations......Page 211 4.6 Schur Complements Again (optional)......Page 215 5.1 The {1}-Inverse......Page 220 5.2 {1,2}-Inverses......Page 229 5.3 Constructing Other Generalized Inverses......Page 231 5.4 {2}-Inverses......Page 238 5.5 The Drazin Inverse......Page 244 5.6 The Group Inverse......Page 251 6.1 The Normed Linear Space C^n......Page 254 6.2 Matrix Norms......Page 265 6.2.1.1 Norms......Page 273 7.1 The Inner Product Space C^n......Page 278 7.2 Orthogonal Sets of Vectors in C^n......Page 283 7.3 QR Factorization......Page 290 7.3.1 Kung's Algorithm......Page 295 7.3.2.1 The QR Factorization......Page 297 7.4 A Fundamental Theorem of Linear Algebra......Page 299 7.5 Minimum Norm Solutions......Page 303 7.6 Least Squares......Page 306 8.1 Orthogonal Projections......Page 312 8.2 The Geometry of Subs paces and the Algebra of Projections......Page 320 8.3 The Fundamental Projections of a Matrix......Page 330 8.4 Full Rank Factorizations of Projections......Page 334 8.5 Affine Projections......Page 336 8.6 Quotient Spaces (optional)......Page 345 9.1 Eigenstuff......Page 350 9.1.1.1 Eigenvalues and Eigenvectors in MATLAB......Page 358 9.2 The Spectral Theorem......Page 359 9.3 The Square Root and Polar Decomposition Theorems......Page 368 10.1 Diagonalization with Respect to Equivalence......Page 372 10.2 Diagonali.lation with Respect to Similarity......Page 378 10.3 Diagonahzation with Respect to a Unitary......Page 392 10.3.1.1 Schur Triangularization......Page 397 10.4 The Singular Value Decomposition......Page 398 10.4.1.1 The Singular Value Decomposition......Page 406 11.1.1 Jordan Blocks......Page 410 11.1.2 Jordan Segments......Page 413 11.1.2.1 MATLAB Moment......Page 416 11.1.3 Jordan Matrices......Page 417 11.1.3.1 MATLAB Moment......Page 418 11.1.4 Jordan's Theorem......Page 419 11.1.4.1 Generalized Eigenvectors......Page 423 11.2 The Smith Normal Form (optional)......Page 443 12.1 Bilinear Forms......Page 452 12.2 Matrices Associated to Bilinear Forms......Page 458 12.3 Orthogonality......Page 461 12.4 Symmetric Bilinear Forms......Page 463 12.5 Congruence and Symmetric Matrices......Page 468 12.6 Skew-Symmetric Bilinear Forms......Page 471 12.7 Tensor Products of Matrices......Page 473 12.7.1.1 Tensor Product of Matrices......Page 477 A.1 What Is a Scalar?......Page 480 A.2 The System of Complex Numbers......Page 485 A.3.1.4 Commutative Law of Addition......Page 487 A.3.1.9 Existence of Inverses......Page 488 A.4 Complex Conjugation, Modulus, and Distance......Page 489 A.4.2 Basic Facts about Magnitude......Page 490 A.4.3 Basic Properties of Distance......Page 491 A.5 The Polar Form of Complex Numbers......Page 494 A.6 Polynomials over C......Page 501 A.7 Postscript......Page 503 B.1 Introduction......Page 506 B.2 Matrix Addition......Page 508 B.3 Scalar Multiplication......Page 510 B.4 Matrix Multiplication......Page 511 B.5 Transpose......Page 516 B.5.1.1 Matrix Manipulations......Page 523 B.6 Submatrices......Page 524 B.6.1.1 Getting at Pieces of Matrices......Page 527 C.1 Motivation......Page 530 C.2 Defining Determinants......Page 533 C.3.2 The Cauchy-Binet Theorem......Page 538 C.3.3 The Laplace Expansion Theorem......Page 541 C.4 The Trace of a Square Matrix......Page 549 D.1 Spanning......Page 552 D.2 Linear Independence......Page 554 D.3 Basis and Dimension......Page 555 D.4 Change of Basis......Page 559 Index......Page 564 Highlighting The Generalized Inverse Of A Matrix And The Method Of Full-rank Factorization, Matrix Theory: From Generalized Inverses To Jordan Form Probes Introductory As Well As More Sophisticated Linear Algebra Concepts. This Presentation Helps Connect Linear Algebra To More Advanced Abstract Algebra And Matrix Theory.--jacket. Idea Of Inverse -- Generating Invertible Matrices -- Subspaces Associated To Matrices. -- The Moore-penrose Inverse -- Generalized Inverses -- Norms -- Inner Products -- Projections -- Spectral Theory -- Matrix Diagonalization -- Jordan Canonical Form -- Multilinear Matters -- Appendix A: Complex Numbers -- Appendix B: Basic Matrix Operations -- Appendix C: Determinants -- Appendix D: A Review Of Basics. Idea Of Inverse Systems Of Linear Equationsthe Special Case Of Square Systemsgenerating Invertible Matricesa Brief Review Of Gauss Elimination With Back Substitutionelementary Matricesthe Lu And Ldu Factorizationthe Adjugate Of A Matrixthe Frame Algorithm And The Cayley-hamilton Theoremsubspaces Associated To Matricesfundamental Subspacesa Deeper Look At Rankdirect Sums And Idempotentsthe Index Of A Square Matrixleft And Right Inversesthe Moore-penrose Inverserow Reduced Echelon Form And Matrix Equivalencethe Hermite Echelon Formfull Rank Factorizationthe Moore-penrose Inversesolving Systems Of Linear Equationsschur Complements Againgeneralized Inversesthe {1}-inverse{1,2}-inversesconstructing Other Generalized Inverses{2}-inversesthe Drazin Inversethe Group Inversenormsthe Normed Linear Space Cnmatrix Normsinner Productsthe Inner Product Space Cnorthogonal Sets Of Vectors In Cnqr Factorizationa Fundamental Theorem Of Linear Algebraminimum Norm Solutionsleast^ Squaresprojectionsorthogonal Projectionsthe Geometry Of Subspaces And The Algebra Of Projectionsthe Fundamental Projections Of A Matrixfull Rank Factorizations Of Projectionsaffine Projectionsquotient Spacesspectral Theoryeigenstuffthe Spectral Theoremthe Square Root And Polar Decomposition Theoremsmatrix Diagonalizationdiagonalization With Respect To Equivalencediagonalization With Respect To Similaritydiagonalization With Respect To A Unitarythe Singular Value Decompositionjordan Canonical Formjordan Form And Generalized Eigenvectorsthe Smith Normal Formmultilinear Mattersbilinear Formsmatrices Associated To Bilinear Formsorthogonalitysymmetric Bilinear Formscongruence And Symmetric Matricesskew-symmetric Bilinear Formstensor Products Of Matricesappendix A: Complex Numberswhat Is A Scalar?the System Of Complex Numbersthe Rules Of Arithmetic In Ccomplex Conjugation, Modulus,^ And Distancethe Polar Form Of Complex Numberspolynomials Over Cpostscriptappendix B: Basic Matrix Operationsintroductionmatrix Additionscalar Multiplicationmatrix Multiplicationtransposesubmatricesappendix C: Determinantsmotivationdefining Determinantssome Theorems About Determinantsthe Trace Of A Square Matrixappendix D: A Review Of Basicsspanninglinear Independencebasis And Dimensionchange Of Basisindex. Robert Piziak, P.l. Odell. Includes Bibliographical References And Index. Suitable for the second-semester course in linear algebra, this work creates a bridge from linear algebra concepts to more advanced abstract algebra and matrix theory. It focuses on the development of the Moore-Penrose inverse. It provides MATLAB[registered] examples and exercises as well as homework problems and suggestions for further reading.