406 INDEX

inverses and determinants, 97invertible, 48invertible matrix

product of elementary matrices, 120irreducible, 196

relatively prime, 197iterative methods

alternate proof of convergence, 351convergence criterion, 346diagonally dominant, 351proof of convergence, 349

Jocobi method, 342Jordan block, 245, 247Jordan canonical form

existence and uniqueness, 247powers of a matrix, 249

ker, 121kernel, 59kernel of a product

direct sum decomposition, 235Krylov sequence, 239

Lagrange form of remainder, 170Laplace expansion, 96least squares, 127, 283least upper bound, 19linear combination, 39, 60, 94linear transformation, 54, 215

defined on a basis, 216dimension of vector space, 216existence of eigenvector, 229kernel, 233matrix, 55minimal polynomial, 229rotation, 57

linear transformationsa vector space, 215commuting, 235composition, matrices, 224sum, 215, 281

linearly dependent, 60linearly independent, 60, 190linearly independent set

extend to basis, 194LU factorization

justification for multiplier method, 133multiplier method, 129solutions of linear systems, 131

main diagonal, 84

Markov matrix, 263limit, 266regular, 266steady state, 263, 266

mathematical induction, 20matrices

commuting, 297notation, 38transpose, 47

matrix, 37differentiation operator, 218injective, 65inverse, 48left inverse, 98lower triangular, 84, 99Markov, 263non defective, 164normal, 164polynomial, 110rank and existence of solutions, 122rank and nullity, 121right and left inverse, 65right inverse, 98right, left inverse, 98row, column, determinant rank, 99self adjoint, 157stochastic, 263surjective, 65symmetric, 156unitary, 160upper triangular, 84, 99

matrixpositive definite, 323

matrix exponential, 352matrix multiplication

definition, 40entries of the product, 42not commutative, 41properties, 46vectors, 39

matrix of linear transformationorthonormal bases, 221

migration matrix, 266minimal polynomial, 81, 228, 236

eigenvalues, eigenvectors, 229finding it, 251generalized eigenspaces, 237

minimal polynomialalgebraic number, 205

minor, 83, 96mixed partial derivatives, 169