Adjoint matrices, defined, 87, 98; see also Hermitian matrices
Bandwidths of matrices, 208–15
Bilinear form, 104
Binomial coefficient, asymptotic form for, 227
Biorthogonality, principle of, 187
Canonical diagonalization of Hermitian matrices, 100
Canonical forms:
diagonal, 75–79
numerical instability of, 232–33
Cauchy-Schwarz inequality, 84–85, 165
Cayley-Hamilton theorem, 113–14, 127–31
Characteristic equation, 68, 71–72
Column expansion, 10–13
Column vectors, defined, 2
Companion matrix, 75
Complex numbers, as field, 27–30, 32
Condition-number of a matrix, 174–77
defined, 175
Conservation of energy, law of, 104–5
Convex bodies, vector norms and, 171–73
Courant minimax theorem, 146–18
Derivative of a determinant, 23–24
Determinants, 1–24
defined, 3–4
derivatives of, 23–24
evaluation of rank by, 43–47
of matrix products, 20–23
properties of, 5–9
Vandermonde’s, 12–13
as volumes, 153–57
Diagonal canonical form, 75–79
Diagonalization:
canonical, of Hermitian matrices, 100
simultaneous, of two quadratic forms, 103–8
Differential equations, matrix analysis of, 57–70
Dimension of a linear space, 33–37
defined, 35
rank as, 40
Disjoint components of Gershgorin circles, 162
Eigensolutions, 67–70
Eigenvalues:
basis for theory of, 38
computation of eigenvectors from, 228–31
defined, 71
dominant:
and computation of smaller eigenvalues, 241–50
defined, 235
method of iteration for, 235–39
in left-hand plane, 268–78
matrices with distinct, 71–74
QR method of computing, 279, 283–89
of similar matrices, 76–77
solution of differential equations by, 66–70
trace as sum of, 80–82
of tridiagonal and Hessenberg matrices, 251–56
variational properties of, 141–92
continuous dependence on matrices, 191–92
Eigenvectors:
basis for theory of, 38
computation of, from known eigenvalues, 228–31
defined, 71
principal axes of ellipsoids as, 95–98
solution of differential equations by, 66–70
of tridiagonal and Hessenberg matrices, 251–56
variational properties of, 141–92
Elimination, method of, 195–202
Ellipsoids, principal axes of, 94–98
Energy, law of conservation of, 104–5
Equal matrices, 14–15
Errors:
in computed eigenvalues and eigenvectors, 141–92
rounding, reduction of, 201–2, 217–21
Euclidean space, 32
rigid motions in, 117–20
Even permutation, defined, 6
Expansions, row and column, 10–13
Exponential matrix, solution of differential equations by, 64–65
Factorization by triangular matrices, 203–8
Fibonacci numbers, 80
Field, defined, 26–27
Finite-dimensional linear vector space, 32
basis of, 89–93
Francis and Kublanovskaya, method of (QR method), 279, 283–89
Fredholm theory of integral equations, 153
Functional analysis, 32
Gauss-Seidei method, 221–27
Generalized unit sphere, defined, 171
Gershgorin’s theorem, 161–63
Sturm’s theorem with, 255–56
Gram-Schmidt orthogonalization process, 89–93, 279
Guillemin, E. A., 270
Hadamard’s inequality, 153–57
Hermitian matrices, 98–102, 110
commuting, 115–17
factorization and, 206–8
Hessenberg, eigenvalues and eigenvectors of, 251–58
positive-definite, 105–8
determinant criterion for, 152–53
Gauss-Seidel method and, 224–27
Rayleigh principle and, 141–45
Weyl’s theorem and, 157–61
Hessenberg matrices, eigenvalues and eigenvectors of, 251–58
Homogeneous systems:
defined, 26
solvability of, 38–42
Householder and Bauer, method of, 251, 258–67
Hurwitz, 268
Inclusion principle, 149–51
as determinant criterion for positive definiteness, 152–53
Inequalities:
Hadamard’s, 153–57
Weyl’s, 157–61
Inhomogeneous systems:
defined, 26
fundamental solution of, 60
general m × n, 48–49
reduction of, by matrix analysis, 60–62
Inner products, defined, 84, 89, 98
Invariants, 80–82
Inverse matrix, 17–19
Irreducible matrices, 181–84
Iterative methods, 221–27
for dominant eigenvalues, 235–39
Jordan canonical form, 68, 121–39
numerical instability of, 232–33
proof of, 132–39
Keller’s theorem, 224–27
Kronecker delta, 18
Laguerre polynomials, 93
Least-squares solution, 50–55
Linear combinations, 33–37
Linear equations:
differential, matrix analysis of, 57–70
direct solution of large systems of, 208–15
theory of, 26–55
Linear space, see Linear vector space
Linear vector space, 26–32
basis of, 33–37
of mutually orthogonal unit vectors, 89–93
dimension of, 33–37
rank as dimension, 40
finite-dimensional, 32
basis of, 89–93
Linearly dependent and independent vectors, 33–37
Lyapunov’s theorem, 270–72
Mass-spring systems, 103–8
Matrix, defined, 2
Matrix norms, 65
related, 167–71
defined, 167–68
Matrix products, determinants of, 20–23
Metric, Riemannian, 164
Minimax theorem, Courant, 146–48
Multiplication of matrices, 15–16, 87
Newton’s method, 256
Nonsingular matrix, defined, 23
Normal equations, 51–55
defined, 51
Normal matrices, 115–20
defined, 115
Norms, vector, 163–73
convex bodies and, 171–73
regular, defined, 164
defined, 39
Numbers as fields, 26–32
Numerical instability of the Jordan canonical form, 232–33
Numerical methods, 194–289
computation of eigenvectors, 228–31
direct solution of large systems of linear equations, 208–15
elimination, 195–202
factorization by triangular matrices, 203–8
identification of stable matrices, 268–78
iterative methods, 221–27
for dominant eigenvalues, 235–39
to obtain the smaller eigenvalues, 241–50
method of Householder and Bauer, 251, 258–67
reduction of rounding error, 201–2, 217–21
with tridiagonal and Hessenberg matrices, 251–58
Odd permutation, defined, 6
Orthogonality:
defined, 89
Gram-Schmidt process for, 89–93
of principal axes of ellipsoids, 95–98
Parallelograms, Hadamard’s inequality and, 154–56
Permutation:
even, defined, 6
odd, defined, 6
sign of the, 4
Perron’s theorem, 177–82
Perturbation theory, 141–92
described, 141
Pivotal element, 201–2
Positive and irreducible matrices, 177–84
Positive-definite matrices, 105–8
determinant criterion for, 152–53
Gauss-Seidel method and, 224–27
Principal axes of ellipsoids, 94–98
Principal vectors, 125–31
defined, 125–26
Rank of a matrix, 49
defined, 40–41
evaluation of, by determinants, 43–47
Rational numbers, as field, 26
Rayleigh principle, 141–45, 146
Real numbers, as field, 26, 28, 30, 31, 32
Related matrix-norms, 167–71
defined, 167–68
Riemannian metric, 164
Rigid motions in Euclidian space, 117–20
Rotation, theory of normal matrices and, 117–20
Rounding error:
in method of iteration for dominant eigenvalues, 239
Routh’s theorem, 268–70
Row expansion, 10–13
Row vectors, defined, 2
Rutishauser’s method, 289
Scalars, defined, 27
Schwarz inequality, see Cauchy-Schwarz inequality
Sign of the permutation, 4
Similar matrices:
defined, 75
eigenvalues of, 76–77
Singular matrix, defined, 23
Skew-Hermitian matrices, 115–17
Space, see Euclidean space and Null space
Spectrum, perturbations of, 186–89
Stable matrices, numerical identification of, 268–78
Sturm’s theorem, 254–56
Submatrices:
defined, 43
evaluation of rank and, 43–47
Successive over-relaxation, 227
Symmetric matrices, 94–95, 102
mass and spring matrices as, 105
Trace, 80–82
defined, 80
Transpose of a matrix, formation of, 6
Triangular matrices:
accurate unitary reduction to, 279–83
factorization by, 203–8
recursion solution of, 195
Triangularization, unitary, 109–14
Tridiagonal matrices, eigenvalues and eigenvectors of, 251–58
Uncoupling property of canonical diagonal form, 78–79
Unit vectors:
defined, 89
mutually orthogonal, 89–93
accurate reduction of, to triangular form, 279–83
defined, 87
as normal matrices, 115–17
Unitary triangularization, 109–14
Unsolvablc systems, least-squares solution of, 50–55
Vandermonde’s determinant, 12–13
Varga, R. S., 227
Variational principles, 141–92
Vector addition, 27–28
Vector norms, 163–73
convex bodies and, 171–73
regular, defined, 164
Vector space, see Linear vector space
Vectors, see also Eigenvectors
defined, 2
length of, and unitary matrices, 83–87
linearly dependent and independent, 33–37
matrices and, 14–16
principal, 125–31
defined, 125–26
unit:
defined, 89
mutually orthogonal, 89–93
Volumes, determinants as, 153–57
Weyl’s inequalities, 157–61