482 INDEX

right inverse, 95right polar factorization, 314rotations

about given vector, 447row and column space, 156row equivalent, 150row operations, 58, 112, 143row rank, 155row reduced echelon form, 59, 148

existence, 148uniqueness, 150

row space, 155

scalar product, 31scalars, 16, 81scaling factor, 347Schur’s theorem, 306set notation, 1sgn, 129

uniqueness, 130shifted inverse power method, 349sign of a permutation, 130similar matrices, 442similarity

block diagonal matrix, 433upper triangular block diagonal, 435

similarity and equivalence, 442similarity relation, 261similarity transformation, 442simple field extension, 395simplex tableau, 222, 224simultaneous corrections, 333singular value decomposition, 317singular values, 317skew lines, 53skew symmetric, 92, 102slack variables, 222, 225solution space, 195span, 134, 148, 376spanning sets, 165spectrum, 248speed, 27splitting field, 391strictly upper triangular, 467Sturm Liouville problem, 418subspace, 164

has a basis, 381surjective, 425

Sylvester’s theorem, 427symmetric, 92, 102symmetric matrix, 291

trace, 330sum of eigenvalues, 330

transposedot product, 173

triangle inequality, 22, 32, 39, 406complex numbers, 5

trigonometrysum of two angles, 189

union, 1unitary, 306upper Hessenberg form, 365

variation of constants formula, 401variational inequality, 421vector

components, 18vector addition

geometric meaning, 19vector space, 373

dimension, 380vector space axioms, 17, 82, 373vectors, 15, 25, 83

column, 83row vector, 84

velocity, 27

well ordered, 2well ordering, 2work, 35Wronskian, 126, 408Wronskian alternative, 401

zero matrix, 82