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Most of the existing monographs on generalized inverses are based on linear algebra tools and geometric methods of Banach (Hilbert) spaces to introduce generalized inverses of complex matrices and operators and their related applications, or focus on generalized inverses of matrices over special rings like division rings and integral domains, and does not include the results in general algebraic structures such as arbitrary rings, semigroups and categories, which are precisely the most general cases. In this book, five important generalized inverses are introduced in these algebraic structures. Moreover, noting that the (pseudo) core inverse was introduced in the last decade and has attracted much attention, this book also covers the very rich research results on it, so as to be a necessary supplement to the existing monographs. This book starts with decompositions of matrices, introduces the basic properties of generalized inverses of matrices, and then discusses generalized inverses of elements in rings and semigroups, as well as morphisms in categories. The algebraic nature of generalized inverses is presented, and the behavior of generalized inverses are related to the properties of the algebraic system. Scholars and graduate students working on the theory of rings, semigroups and generalized inverses of matrices and operators will find this book helpful.

Linear operators --- Generalized inverses. --- Operadors lineals --- Associative rings. --- Associative algebras. --- Group theory. --- Associative Rings and Algebras. --- Group Theory and Generalizations.

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Matrix inversion --- 512.64 --- Inverse matrices --- Inverse of a matrix --- Inversion, Matrix --- Linear operators --- Matrices --- Linear and multilinear algebra. Matrix theory --- Generalized inverses --- Matrix inversion. --- 512.64 Linear and multilinear algebra. Matrix theory

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Analysis of variance --- Matrix inversion --- Analyse de variance --- Matrices --- Inversion --- 519.237 --- #WWIS:IBM/STAT --- Inverse matrices --- Inverse of a matrix --- Inversion, Matrix --- Linear operators --- ANOVA (Analysis of variance) --- Variance analysis --- Mathematical statistics --- Experimental design --- Multivariate statistical methods --- Generalized inverses --- 519.237 Multivariate statistical methods --- Opérateurs linéaires --- Inverses généralisés --- Opérateurs linéaires --- Inverses généralisés

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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.

Algebra --- Matrices --- Algebras, Linear --- Matrix inversion --- 512.643 --- Matrices and linear mappings. Matrix theory. Determinants. Eigenvalues --- 512.643 Matrices and linear mappings. Matrix theory. Determinants. Eigenvalues --- Inverse matrices --- Inverse of a matrix --- Inversion, Matrix --- Linear operators --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Linear algebra --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- Generalized inverses --- Matrices - Textbooks --- Algebras, Linear - Textbooks --- Matrix inversion - Textbooks

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Numerical solutions of algebraic equations --- Mathematical statistics --- Matrices --- 514 --- 519.6 --- 681.3*G13 --- Algebra, Matrix --- Cracovians (Mathematics) --- Matrix algebra --- Matrixes (Algebra) --- Algebra, Abstract --- Algebra, Universal --- Geometry --- Computational mathematics. Numerical analysis. Computer programming --- Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- Matrices. --- 681.3*G13 Numerical linear algebra: conditioning; determinants; Eigenvalues; error analysis; linear systems; matrix inversion; pseudoinverses; sparse and very largesystems --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 514 Geometry --- Linear operators --- Opérateurs linéaires --- Generalized inverses --- Inverses généralisés --- Opérateurs linéaires --- Inverses généralisés