# Matrix Theory : From Generalized Inverses to Jordan Form

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## Description

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.

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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.

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## Product details

- Hardback | 568 pages
- 157.5 x 231.1 x 35.6mm | 884.52g
- 22 Feb 2007
- Taylor & Francis Ltd
- Chapman & Hall/CRC
- Boca Raton, FL, United States
- English
- 27 black & white illustrations, 3 black & white tables
- 1584886250
- 9781584886259

## Review quote

Each chapter ends with a list of references for further reading. Undoubtedly, these will be useful for anyone who wishes to pursue the topics deeper. ... the book has many MATLAB examples and problems presented at appropriate places. ... the book will become a widely used classroom text for a second course on linear algebra. It can be used profitably by graduate and advanced level undergraduate students. It can also serve as an intermediate course for more advanced texts in matrix theory. This is a lucidly written book by two authors who have made many contributions to linear and multilinear algebra. -K.C. Sivakumar, IMAGE, No. 47, Fall 2011 Always mathematically constructive, this book helps readers delve into elementary linear algebra ideas at a deeper level and prepare for further study in matrix theory and abstract algebra. -L'enseignement Mathematique, January-June 2007, Vol. 53, No. 1-2

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## About Robert Piziak

Baylor University, Texas, USA Baylor University, Texas, USA

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## Table of contents

THE IDEA OF INVERSE Solving Systems of Linear Equations The Special Case of "Square" Systems GENERATING INVERTIBLE MATRICES A Brief Review of Gauss Elimination with Back Substitution Elementary Matrices The LU and LDU Factorization The Adjugate of a Matrix The Frame Algorithm and the Cayley-Hamilton Theorem SUBSPACES ASSOCIATED TO MATRICES Fundamental Subspaces A Deeper Look at Rank Direct Sums and Idempotents The Index of a Square Matrix Left and Right Inverses THE MOORE-PENROSE INVERSE Row Reduced Echelon Form and Matrix Equivalence The Hermite Echelon Form Full Rank Factorization The Moore-Penrose Inverse Solving Systems of Linear Equations Schur Complements Again GENERALIZED INVERSES The {1}-Inverse {1,2}-Inverses Constructing Other Generalized Inverses {2}-Inverses The Drazin Inverse The Group Inverse NORMS The Normed Linear Space Cn Matrix Norms INNER PRODUCTS The Inner Product Space Cn Orthogonal Sets of Vectors in Cn QR Factorization A Fundamental Theorem of Linear Algebra Minimum Norm Solutions Least Squares PROJECTIONS Orthogonal Projections The Geometry of Subspaces and the Algebra of Projections The Fundamental Projections of a Matrix Full Rank Factorizations of Projections Affine Projections Quotient Spaces SPECTRAL THEORY Eigenstuff The Spectral Theorem The Square Root and Polar Decomposition Theorems MATRIX DIAGONALIZATION Diagonalization with Respect to Equivalence Diagonalization with Respect to Similarity Diagonalization with Respect to a Unitary The Singular Value Decomposition JORDAN CANONICAL FORM Jordan Form and Generalized Eigenvectors The Smith Normal Form MULTILINEAR MATTERS Bilinear Forms Matrices Associated to Bilinear Forms Orthogonality Symmetric Bilinear Forms Congruence and Symmetric Matrices Skew-Symmetric Bilinear Forms Tensor Products of Matrices APPENDIX A: COMPLEX NUMBERS What is a Scalar? The System of Complex Numbers The Rules of Arithmetic in C Complex Conjugation, Modulus, and Distance The Polar Form of Complex Numbers Polynomials over C Postscript APPENDIX B: BASIC MATRIX OPERATIONS Introduction Matrix Addition Scalar Multiplication Matrix Multiplication Transpose Submatrices APPENDIX C: DETERMINANTS Motivation Defining Determinants Some Theorems about Determinants The Trace of a Square Matrix APPENDIX D: A REVIEW OF BASICS Spanning Linear Independence Basis and Dimension Change of Basis INDEX

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