Elementary Linear Algebra [Lecture notes] - Kenneth L. Kuttler

INDEX 441 relatively prime, 332 inverse, 77 isomorphism, 155 invertible, product of elementary matrices, Jacobi, 282 148 Jacobi method, 282 left inverse, 115 Jordan block, 399 left inverse, right inverse, 162 joule, 30 lower triangular, 92, 116 main diagonal, 220 ker, 164 one to one, onto, 162 kernel, 143, 164 polynomial, 117 Kirchoff’s law, 63, 64 raising to a power, 222 Kroneker delta, 39 right inverse, 115 Kroneker symbol, 76 right inverse left inverse and inverse, 129 rotation, 158 Laplace expansion, 91, 113 rotation about given vector, 159 leading entry, 48 self adjoint, 240, 275 least square approximation, 261 symmetric, 240 Leontief model, 88 transpose, 75 linear combination, 70, 112, 125 upper triangular, 92, 116 linear independence matrix exponential, 223, 224 matrix inverse definition, 135 finding it, 78, 79 enlarging to form a basis, 328 matrix multiplication equivalent conditions, 135 ij entry, 73 linear relationships, 126 properties, 74 finding them, 137 vectors, 70 linear transformation, 155, 365 mean square approximation, 356 defined on a basis, 366 migration matrix, 227 matrix, 157, 163 minimal polynomial, 154, 370 rotation, 156 computation, 394 linear transformations uniqueness, 370 commuting, 368 minimal polynomial dimension, 365 algebraic number, 339 linearly dependent, 324 minimization and orthogonality, 353 linearly independent, 134, 324 minor, 91, 92, 113 linearly independent sets, 138 monic, 331 LU deomposition monic polynomial, 370 non existence, 173 multipliers, 178 LU factorization, 173 by inspection, 173 Neuman series, 88 justification, 176 Newton, 22 multipliers, 174 nilpotent, 103, 375 solving systems, 175 non defective main diagonal, 92 minimal polynomial, 395 Markov matrices, 227 nondefective, 251 math induction, 2 nondefective eigenvalue, 216 mathematical induction, 2 normed linear space, 350 matrices normed vector space, 350 null space, 143, 164 more columns than rows, 127 nullity, 144 multiplication, 71 one to one, onto, 148 one to one, 157 similar, 219 rank, 153 matrix, 67 composition of linear transformations, 383 onto, 157 identity, 76 open ball, 17

442 INDEX operator norm, 289 composition of linear transformations, 383 orthogonal matrix, 103, 167, 180, 243 projection, 29, 150 projection of a vector, 29 switching two unit vectors, 181 projections orthogonal projection, 354 orthogonality and minimization, 261 matrix, 161 orthonormal, 244, 257 QR decomposition, 258 independent, 350 QR factorization, 181, 258 orthonormal set, 350 thin, 258 p norms, 293 quadratic form, 253 parallelepiped, 38 quadratic formula, 8 volume, 38 rank parallelogram identity, 363 column determinant and row, 130 particular solution, 163 existence of solutions, 145 partitioned matrix, 253 finding the rank, 132 permutation, 108 linear transformation, 381 permutation matrices, 119 permutation symbol, 40 rank and singular values, 269 rank of a matrix, 130, 147 reduction identity, 40 Rayleigh quotient, 305 perp, 145 reflection perpendicular, 28 pivot, 53 across a given vector, 167 pivot column, 49, 126 regression line, 263 pivot columns, 49 regular Sturm Liouville problem, 360 pivot position, 49 resultant, 22 pivot positions, 49 right handed system, 34 PLU factorization, 178 right inverse, 79 points and vectors, 13 right polar factorization, 265 polar form complex number, 6 rotations polarization identity, 363 polynomial, 330 about given vector, 383 row and column space, 131 degree, 330 row equivalent, 127 divides, 331 row operations, 48, 93, 119 division, 330 row rank, 130 equal, 330 row reduced echelon form, 48, 125 greatest common divisor, 331 greatest common divisor description, 331 existence, 125 greatest common divisor, uniqueness, 331 uniqueness, 127 irreducible, 331 row space, 129 irreducible factorization, 332 relatively prime, 331 scalar product, 25 root, 330 scalars, 14, 67 polynomial scaling factor, 295 matrix coefficients, 117 Schur’s theorem, 259 polynomials set notation, 1 canceling, 332 sgn, 107 factoring, 7 factorization, 333 uniqueness, 108 position vector, 15 shifted inverse power method, 297 power method, 295 preserving distance, 266 complex eigenvalues, 304 principal axes, 253 sign of a permutation, 108 principal directions, 226 similar matrices, 379 product of matrices similarity block diagonal matrix, 371 upper triangular block diagonal, 373 similarity and equivalence, 380 similarity relation, 220

INDEX 443 similarity transformation, 379 Wronskian, 106, 352 simple field extension, 341 Wronskian alternative, 345 simplex tableau, 188, 189 simultaneous corrections, 282 zero matrix, 68 singular value decomposition, 268 singular values, 268 skew lines, 44 skew symmetric, 76, 84 slack variables, 188, 190 solution space, 164 span, 112, 125, 324 spanning sets, 138 spectrum, 210 speed, 23 splitting field, 337 strictly upper triangular, 399 Sturm Liouville problem, 360 subspace, 137 has a basis, 329 substituting matrix into polynomial identity, 117 surjective, 365 Sylvester’s theorem, 367 symmetric, 76, 84 symmetric matrix, 245 trace, 279 sum of eigenvalues, 279 transpose dot product, 145 triangle inequality, 19, 26, 32, 349 complex numbers, 5 trigonometry sum of two angles, 159 union, 1 unitary, 258 upper Hessenberg form, 313 variation of constants formula, 346 variational inequality, 363 vector components, 16 vector addition geometric meaning, 16 vector space, 321 dimension, 328 vector space axioms, 14, 68, 321 vectors, 13, 22, 69 column, 69 row vector, 69 velocity, 23 well ordered, 2 well ordering, 2 work, 29