Linear Algebra

similar, 318similarity, 347singular, 27skew-symmetric, 305submatrix, 297sum, 207symmetric, 118, 138, 208, 215, 223,262trace, 208, 224, 394transition, 275transpose, 19, 126, 208triangular, 198, 224, 324unit, 217Vandermonde, 305matrix equivalence, 236–243canonical form, 239definition, 238matrix:form, 56meanarithmetic, 44geometric, 44member, A-7method of powers, 395–398minimal polynomial, 215, 376minor, 321morphism, 155multilinear, 298multiplicationmatrix-matrix, 210matrix-vector, 192MuPAD, 62mutual inclusion, A-7natural representative, A-12networks, 72–77Kirchhoff’s Laws, 73nilpotent, 364–374canonical form for, 370definition, 366matrix, 366transformation, 366nilpotentcyindex, 366nonsingular, 202, 226homomorphism, 184matrix, 27normalize, 252nullity, 182nullspace, 182closure of, 363generalized, 363Octave, 62odd functions, 99, 137one-to-one function, A-9onto function, A-9opposite map, 284orderof a recurrence, 402ordered pair, A-8orientation, 315, 318orthogonal, 42basis, 250complement, 257mutually, 249projection, 257orthogonal matrix, 283orthogonalization, 251orthonormal basis, 252orthonormal matrix, 281–286pairordered, A-8parallelepiped, 315parallelogram rule, 34parameter, 13partial pivoting, 70partition, A-10–A-12matrix equivalence classes, 238, 240row equivalence classes, 51partitionsinto isomorphism classes, 164permutation, 301inversions, 307matrix, 220signum, 308permutation expansion, 302, 306, 328perp, 257perpendicular, 42perspectivetriangles, 337Physics problem, 1pivotingfull, 70partialscaled, 70pivoting on rows, 4plane figure, 281congruence, 281point