Matrix AnalysisLinear algebra and matrix theory have long been fundamental tools in mathematical disciplines as well as fertile fields for research. In this book the authors present classical and recent results of matrix analysis that have proved to be important to applied mathematics. Facts about matrices, beyond those found in an elementary linear algebra course, are needed to understand virtually any area of mathematical science, but the necessary material has appeared only sporadically in the literature and in university curricula. As interest in applied mathematics has grown, the need for a text and reference offering a broad selection of topics in matrix theory has become apparent, and this book meets that need. This volume reflects two concurrent views of matrix analysis. First, it encompasses topics in linear algebra that have arisen out of the needs of mathematical analysis. Second, it is an approach to real and complex linear algebraic problems that does not hesitate to use notions from analysis. Both views are reflected in its choice and treatment of topics. |
Contents
Review and miscellanea | 1 |
02 Matrices | 4 |
03 Determinants | 7 |
04 Rank | 12 |
05 Nonsingularity | 14 |
07 Partitioned matrices | 17 |
08 Determinants again | 19 |
09 Special types of matrices | 23 |
53 Algebraic properties of vector norms | 268 |
54 Analytic properties of vector norms | 269 |
55 Geometric properties of vector norms | 281 |
56 Matrix norms | 290 |
57 Vector norms on matrices | 320 |
58 Errors in inverses and solutions of linear systems | 335 |
Location and perturbation of eigenvalues | 343 |
61 Gerśgorin discs | 344 |
010 Change of basis | 30 |
Eigenvalues eigenvectors and similarity | 33 |
11 The eigenvalueeigenvector equation | 34 |
12 The characteristic polynomial | 38 |
13 Similarity | 44 |
14 Eigenvectors | 57 |
Unitary equivalence and normal matrices | 65 |
21 Unitary matrices | 66 |
22 Unitary equivalence | 72 |
23 Schurs unitary triangularization theorem | 79 |
24 Some implications of Schurs theorem | 85 |
25 Normal matrices | 100 |
26 OR factorization and algorithm | 112 |
Canonical forms | 119 |
a proof | 121 |
some observations and applications | 129 |
the minimal polynomial | 142 |
34 Other canonical forms and factorizations | 150 |
35 Triangular factorizations | 158 |
Hermitian and symmetric matrices | 167 |
41 Definitions properties and characterizations of Hermitian matrices | 169 |
42 Variational characterizations of eigenvalues of Hermitian matrices | 176 |
43 Some applications of the variational characterizations | 181 |
44 Complex symmetric matrices | 201 |
45 Congruence and simultaneous diagonalization of Hermitian and symmetric matrices | 218 |
46 Consimilarity and condiagonalization | 244 |
Norms for vectors and matrices | 257 |
51 Defining properties of vector norms and inner products | 259 |
52 Examples of vector norms | 264 |
62 Gerśgorin discs a closer look | 353 |
63 Perturbation theorems | 364 |
64 Other inclusion regions | 378 |
Positive definite matrices | 391 |
71 Definitions and properties | 396 |
72 Characterizations | 402 |
73 The polar form and the singular value decomposition | 411 |
74 Examples and applications of the singular value decomposition | 427 |
75 The Schur product theorem | 455 |
products and simultaneous diagonalization | 464 |
77 The positive semidefinite ordering | 469 |
78 Inequalities for positive definite matrices | 476 |
Nonnegative matrices | 487 |
81 Nonnegative matrices inequalities and generalities | 490 |
82 Positive matrices | 495 |
83 Nonnegative matrices | 503 |
84 Irreducible nonnegative matrices | 507 |
85 Primitive matrices | 515 |
86 A general limit theorem | 524 |
87 Stochastic and doubly stochastic matrices | 526 |
Complex numbers | 531 |
Convex sets and functions | 533 |
The fundamental theorem of algebra | 537 |
Continuous dependence of the zeroes of a polynomial on its coefficients | 539 |
Weierstrasss theorem | 541 |
References | 543 |
Notation | 547 |
| 549 | |
Common terms and phrases
a₁ Ae Mn algebraic B₁ characteristic polynomial coefficients commute complex numbers conclude condition number congruence Consider converges convex Corollary corresponding defined denote diag diagonal matrix diagonalizable direct sum doubly stochastic eigen eigenvalues eigenvector equal equations Euclidean example Exercise factorization finite function Geršgorin given matrix hence Hermitian matrix Hint identity inequality inner product integer inverse irreducible Jordan blocks Jordan canonical form Lemma Let AEM main diagonal entries matrix Ae matrix norm minimal polynomial Mm,n modulus multiplicity node nonnegative matrix nonsingular matrix nonzero vector normal matrices orthonormal permutation matrix positive definite positive semidefinite Problem Proof rank real numbers real orthogonal real symmetric result row and column scalar sequence Show similar singular value decomposition spectral norm suppose symmetric matrix Theorem tion unit ball unitarily equivalent unitarily invariant norm unitary matrix upper triangular matrix vector norm vector space x*Ax zero