INDEX 515

row rank, 183row reduced echelon form, 41

description, 86unique, 87

scalars, 77Schroder Bernstein theorem, 4Schur’s theorem, 329self adjoint, 315, 363self adjoint nonnegative

roots, 354, 362separable

polynomial, 223separable metric space

Lindeloff property, 252separated sets, 259sequence, 246

Cauchy, 248connecting, 475exact, 474subsequence, 247

sequential compactness, 283sequentially compact, 283sequentially compact set, 253set notation, 1sgn, 171

uniqueness, 173shifted inverse power method, 415

complex eigenvalues, 418short and long exact sequences, 479short exact sequence, 154sign of a permutation, 173similar

matrix and its transpose, 167similar matrices, 107, 189similarity

characteristic polynomial, 189determinant, 189trace, 189

similarity transformation, 107simple field extension, 72simple groups, 231simplex

singular, 456simultaneously diagonalizable, 359

commuting family, 361singular simplex

boundary, 456

singular value decomposition, 372singular values, 372skew symmetric, 81solution set, 39solvable by radicals, 235solvable group, 232span, 53, 176spectral mapping theorem, 304spectral norm, 381spectral radius, 390splitting field, 67, 68

dimension, 68splitting fields

isomorphic, 216normal extension, 221

stochastic matrix, 285subdivision and boundary, 469subdivision map, 467submodule, 152subsequence, 247subspace, 56

complementary, 192vector space, 56

subspacesdirect sum, 119direct sum, basis, 119

substituting matrix into polynomial identity,191

Sylvesterlaw of inertia, 346dimention of kernel of product, 124

Sylvester’s equation, 326symmetric, 81symmetric polynomial theorem, 196symmetric polynomials, 195

the space AU, 326Tietze extension theorem, 268, 446totally bounded, 253totally ordered, 503trace, 117, 167

eigenvalues, 117, 167product, 117, 167similar matrices, 117, 167sum of eigenvalues, 341

transpose, 80properties, 80

transposition, 228