Matrix Theory From Generalized Inverses to Jordan Form

Designed for a second-semester course in linear algebra, this classroom-tested text 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 and uses concrete examples to make arguments clear. The authors provide MATLAB examples and exercises throughout as well as numerous homework problems and suggestions for further reading. The appendices present basic linear algebra topics and related prerequisites. Delving into elementary linear algebra ideas at a deeper level, this book prepares readers for further study in matrix theory and abstract algebra.

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.

THE IDEA OF INVERSESolving Systems of Linear EquationsThe Special Case of "Square" SystemsGENERATING INVERTIBLE MATRICES A Brief Review of Gauss Elimination with Back SubstitutionElementary MatricesThe LU and LDU Factorization The Adjugate of a Matrix The Frame Algorithm and the Cayley-Hamilton TheoremSUBSPACES ASSOCIATED TO MATRICES Fundamental SubspacesA Deeper Look at RankDirect Sums and Idempotents The Index of a Square Matrix Left and Right InversesTHE MOORE-PENROSE INVERSE Row Reduced Echelon Form and Matrix EquivalenceThe Hermite Echelon FormFull Rank Factorization The Moore-Penrose Inverse Solving Systems of Linear EquationsSchur Complements Again GENERALIZED INVERSES The {1}-Inverse {1,2}-Inverses Constructing Other Generalized Inverses {2}-Inverses The Drazin Inverse The Group InverseNORMS The Normed Linear Space CnMatrix NormsINNER PRODUCTS The Inner Product Space CnOrthogonal Sets of Vectors in Cn QR FactorizationA Fundamental Theorem of Linear Algebra Minimum Norm Solutions Least SquaresPROJECTIONSOrthogonal ProjectionsThe Geometry of Subspaces and the Algebra of ProjectionsThe Fundamental Projections of a Matrix Full Rank Factorizations of ProjectionsAffine Projections Quotient SpacesSPECTRAL THEORYEigenstuff The Spectral TheoremThe Square Root and Polar Decomposition TheoremsMATRIX DIAGONALIZATION Diagonalization with Respect to EquivalenceDiagonalization with Respect to SimilarityDiagonalization with Respect to a UnitaryThe Singular Value DecompositionJORDAN CANONICAL FORM Jordan Form and Generalized EigenvectorsThe Smith Normal FormMULTILINEAR MATTERSBilinear FormsMatrices Associated to Bilinear FormsOrthogonality Symmetric Bilinear Forms Congruence and Symmetric Matrices Skew-Symmetric Bilinear FormsTensor Products of MatricesAPPENDIX A: COMPLEX NUMBERSWhat is a Scalar?The System of Complex Numbers The Rules of Arithmetic in CComplex Conjugation, Modulus, and Distance The Polar Form of Complex Numbers Polynomials over C PostscriptAPPENDIX B: BASIC MATRIX OPERATIONS IntroductionMatrix AdditionScalar MultiplicationMatrix MultiplicationTransposeSubmatricesAPPENDIX C: DETERMINANTS MotivationDefining Determinants Some Theorems about Determinants The Trace of a Square MatrixAPPENDIX D: A REVIEW OF BASICSSpanningLinear IndependenceBasis and DimensionChange of BasisINDEX

Piziak, Robert; Odell, P.L.