INDEX 405

division of real numbers, 21Dolittle’s method, 130domain, 10dot product, 33dyadics, 216dynamical system, 158

eigenspace, 145, 237eigenvalue, 81, 143eigenvalues, 102, 174, 228

AB and BA, 106eigenvector, 81, 143eigenvectors

distinct eigenvalues independence, 149elementary matrices, 111empty set, 9equality of mixed partial derivatives, 170equivalence class, 199, 220equivalence of norms, 328equivalence relation, 199, 219Euclidean algorithm, 21exchange theorem, 61existence of a fixed point, 349

fieldordered, 11

field axioms, 11field extension, 200

dimension, 202finite, 202

field extensions, 201finite dimensional normed linear space

completeness, 327equivalence of norms, 328

Foucalt pendulum, 72Fourier series, 286Fredholm alternative, 123, 284free variable, 29Frobenius

inner product, 184Frobenius norm, 315

singular value decomposition, 315Frobinius norm, 321functions, 10fundamental theorem of algebra

plausibility argument, 16rigorous proof, 17

fundamental theorem of arithmetic, 24

Gauss Jordan method for inverses, 49Gauss Seidel method, 344Gelfand, 336

generalized eigenspace, 81generalized eigenspaces, 237, 246generalized eigenvectors, 247Gerschgorin’s theorem, 173Gram Schmidt procedure, 141, 160, 276Gram Schmidt process, 276Gramm Schmidt process, 160Grammian determinant, 293greatest common divisor, 22, 196

characterization, 22greatest lower bound, 19Gronwall’s inequality, 354

Hermitian, 164orthonormal basis eigenvectors, 301positive definite, 304real eigenvalues, 166

Hermitian matrixfactorization, 272positive part, 400positive part, Lipschitz continuous, 400

Hermitian operator, 279largest, smallest, eigenvalues, 302spectral representation, 300

Hessian matrix, 171Holder’s inequality, 331Householder

reflection, 137Householder matrix, 136

idempotent, 76inconsistent, 28initial value problem

uniqueness, 354injective, 10inner product, 33, 273inner product space, 273

adjoint operator, 279parallelogram identity, 274triangle inequality, 274

integers mod a prime, 212integral

operator valued function, 354vector valued function, 353

intersection, 9intervals

notation, 9invariant, 298

subspace, 237invariant subspaces

direct sum, block diagonal matrix, 238