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. 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.