Topic
Matrix analysis
About: Matrix analysis is a research topic. Over the lifetime, 3108 publications have been published within this topic receiving 154539 citations.
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Book•
12 Jan 2005
TL;DR: A review of elementary matrix algebra can be found in this article, with a focus on matrix multiplication and matrix factorizations and Martrix Norms, as well as generalized inverses.
Abstract: Preface. 1. A Review of Elementary Matrix Algebra. 2. Vector Spaces. 3. Eigenvalues and Eigenvectors. 4. Matrix Factorizations and Martrix Norms. 5. Generalized Inverses. 6. Systems of Linear Equations. 7. Partitioned Matrices. 8. Special Matrices and Matrix Operations. 9. Matrix Derivatives and Related Topics. 10. Some Special Topics Related to Quadratic Forms. References. Index.
790 citations
Book•
01 Jan 1985
TL;DR: A survey of Scalar Polynomials can be found in this article, where the Jordan Canonical Form is used to define the normal form of matrix polynomials and normal forms.
Abstract: Maxtrix Algebra. Determinants, Inverse Matrices, and Rank. Linear, Euclidean, and Unitary Spaces. Linear Transformations and Matrices. Linear Transformations in Unitary Spaces and Simple Matrices. The Jordan Canonical Form: A Geometric Approach. Matrix Polynomials and Normal Forms. The Variational Method. Functions of Matrices. Norms and Bounds for Eigenvalues. Perturbation Theory. Linear Matrix Equations and Generalized Inverses. Stability Problems. Matrix Polynomials. Nonnegative Matrices. Appendix 1. A Survey of Scalar Polynomials. Appendix 2. Some Theorems and Notions from Analysis. Appendix 3. Suggestions for Further Reading. Index.
748 citations
Book•
25 Dec 2012
TL;DR: In this article, the second edition of the Second Edition of the first edition, the authors presented a list of symbols for elementary linear and multilinear algebra, including square matrices, tensor and exterior products, with real or complex entries.
Abstract: Preface to the Second Edition.- Preface to the First Edition.- List of Symbols.- 1 Elementary Linear and Multilinear Algebra.- 2 What Are Matrices.- 3 Square Matrices.- 4 Tensor and Exterior Products.- 5 Matrices with Real or Complex Entries.- 6 Hermitian Matrices.- 7 Norms.- 8 Nonnegative Matrices.- 9 Matrices with Entries in a Principal Ideal Domain Jordan Reduction.- 10 Exponential of a Matrix, Polar Decomposition, and Classical Groups.- 11 Matrix Factorizations and Their Applications.- 12 Iterative Methods for Linear Systems.- 13 Approximation of Eigenvalues.- References.- Index of Notation.- General Index.- Cited Names.-
692 citations
TL;DR: In this paper, the structural mechanics of assemblies of bars and pinjoints, particularly where they are simultaneously statically and kinematically indeterminate, are investigated, and an algorithm is set up which determines the rank of the matrix and the bases for the four subspaces.
683 citations
TL;DR: A method is proposed for reducing large matrices by constructing a matrix of lower order which has the same dominant eigenvalues and eigenvectors as the original system.
Abstract: Often it is possible to represent physical systems by a number of simultaneous linear differential equations with constant coefficients, \dot{x} = Ax + r but for many processes (e.g., chemical plants, nuclear reactors), the order of the matrix A may be quite large, say 50×50, 100×100, or even 500×500. It is difficult to work with these large matrices and a means of approximating the system matrix by one of lower order is needed. A method is proposed for reducing such matrices by constructing a matrix of lower order which has the same dominant eigenvalues and eigenvectors as the original system.
614 citations