480 INDEX

inverses and determinants, 117, 137invertible, 92irreducible, 384

relatively prime, 385isomorphism, 185

Jacobi, 333Jacobi method, 333Jordan block, 467joule, 36

ker, 195kernel, 170, 195Kirchoff’s law, 77, 78Kroneker delta, 48Kroneker symbol, 92

Laplace expansion, 110, 136leading entry, 58least square approximation, 309Leontief model, 106linear combination, 80, 84, 134, 148linear independence

definition, 161enlarging to form a basis, 381equivalent conditions, 162

linear relation, 80linear relationships, 150

finding them, 163linear transformation, 185, 425

defined on a basis, 426matrix, 187, 194rotation, 186

linear transformationscommuting, 429dimension, 425

linearly dependent, 376linearly independent, 160, 376linearly independent sets, 165LU deomposition

non existence, 205LU factorization, 205

by inspection, 205justification, 208multipliers, 206solving systems, 208

main diagonal, 111Markov matrices, 269

math induction, 2mathematical induction, 2matrices

more columns than rows, 151multiplication, 86one to one, onto, 176similar, 260

matrix, 81identity, 92inverse, 92invertible, product of elementary ma-

trices, 177left inverse, 138left inverse, right inverse, 193lower triangular, 110, 139main diagonal, 261one to one, onto, 193raising to a power, 263right inverse, 138right inverse left inverse and inverse,

152rotation, 188rotation about given vector, 190self adjoint, 285, 326symmetric, 285transpose, 91upper triangular, 110, 139

matrix exponential, 264, 265matrix inverse

finding it, 94, 95matrix multiplication

ij entry, 88properties, 90vectors, 85

mean square approximation, 412migration matrix, 269minimal polynomial, 184, 431

computation, 460uniqueness, 431

minimal polynomialalgebraic number, 393

minimization and orthogonality, 410minor, 109, 110, 136monic, 384monic polynomial, 431multipliers, 210

Neuman series, 106