514 INDEX

volume, 319partial fractions, 25

decomposition, 26unique, 26

partial order, 503partially ordered set, 503partitioned matrix, 88path components, 460Penrose conditions, 379permutation, 173permutation matrices, 91, 227permutations

cycle, 228Perron’s theorem, 293piecewise continuous, 451pointwise convergence, 258polar form complex number, 11Polish space, 252polynomial, 21

addition, 21degree, 21divides, 22division, 21equality, 21greatest common divisor, 22greatest common divisor, uniqueness,

22irreducible, 22irreducible factorization, 23multiplication, 21relatively prime, 22

polynomialleading term, 21matrix coefficients, 191monic, 21

polynomialscoefficients in a field, 63factoring, 13factorization, 24, 126relatively prime?, 44

polynomials in finitely many algebraic num-bers, 70

positive, 363positive definite

postitive eigenvalues, 371principle minors, 371

positive definite matrix, 386positive self adjoint

products, 363roots, 364

postitive definite, 370power method, 413powers of a matrix

existence of a limit, 285Jordan form, 285stochastic matrix, 285

prime number, 19principal submatrix, 350principle ideal domain, 169principle minors, 371projection map

convex set, 324

QR algorithm, 339, 423convergence, 426convergence theorem, 426non convergence, 340, 430

QR factorization, 316quadratic form, 333quadratic formula, 13quotient group, 222quotient module, 153quotient space, 64, 74quotient vector space, 75

ranknumber of pivot columns, 87

rank of a matrix, 87, 183rank one transformations, 103rational canonical form, 161Rayleigh quotient, 420

how close?, 420regression line, 317regular Sturm Liouville problem, 322relatively prime, 19residue class

integers, 27modulo a prime, 28

retract, 466retraction, 466

deformation, 466Riesz representation theorem, 313right polar factorization, 351, 352ring

including 1, 170row operations, 41, 91