Matrices: Theory and ApplicationsIn this book, Denis Serre begins by providing a clean and concise introduction to the basic theory of matrices. He then goes on to give many interesting applications of matrices to different aspects of mathematics and also other areas of science and engineering. With forty percent new material, this second edition is significantly different from the first edition. Newly added topics include: • Dunford decomposition, • tensor and exterior calculus, polynomial identities, • regularity of eigenvalues for complex matrices, • functional calculus and the Dunford–Taylor formula, • numerical range, • Weyl's and von Neumann’s inequalities, and • Jacobi method with random choice. The book mixes together algebra, analysis, complexity theory and numerical analysis. As such, this book will provide many scientists, not just mathematicians, with a useful and reliable reference. It is intended for advanced undergraduate and graduate students with either applied or theoretical goals. This book is based on a course given by the author at the École Normale Supérieure de Lyon. |
Contents
| 1 | |
Chapter 2 What Are Matrices | 15 |
Chapter 3 Square Matrices | 31 |
Chapter 4 Tensor and Exterior Products | 69 |
Chapter 5 Matrices with Real or Complex Entries | 83 |
Chapter 6 Hermitian Matrices | 109 |
Chapter 7 Norms | 126 |
Chapter 8 Nonnegative Matrices | 149 |
Chapter 10 Exponential of a Matrix Polar Decomposition and Classical Groups | 183 |
Chapter 11 Matrix Factorizations and Their Applications | 207 |
Chapter 12 Iterative Methods for Linear Systems | 225 |
Chapter 13 Approximation of Eigenvalues | 247 |
| 277 | |
| 279 | |
| 283 | |
| 288 | |
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Common terms and phrases
algebraic algorithm associated assume basis bilinear bistochastic block-diagonal block-triangular blockwise canonical characteristic polynomial column companion matrix compute conjugate convex Corollary Deduce detM diagonal blocks diagonal entries diagonalizable dimension domain eigenvalues eigenvector element equal equivalent Euclidean Exercise exists finite formula Gauss–Seidel given GLn(C GLn(k Hence Hermitian matrices Hessenberg matrix homeomorphic HPDn implies induced norm induction inequality integer invariant invertible irreducible isomorphism iteration Jacobi method Lemma Let us define linear form linear map linear subspace LU factorization matrix norm minimal polynomial Mn(A Mn(C Mn(K Mn(R modulus multiplicity nonnegative nonsingular nonzero normal obtain operations orthogonal permutation permutation matrix positive positive-definite positive-semidefinite Proof Proposition prove real numbers respectively satisfies sequence Show similarity invariants spectrum Springer Science+Business Media square matrix symmetric symmetric matrix tensor Theorem triangular tridiagonal unique unitarily unitary matrices upper-triangular vanishes vector space zero