Index

abelian group, 1 cokernel of group diagonal matrices, 232 adjacency matrix, 53 homomorphism, 99 diagonalizable matrix, 232 adjoint matrix, 89 column spaces, 21 diagonalization, 83 affine spaces, 60 columns of a matrix, 279 diagonally dominant, 155 algebra, 11 commutative group, 32 differential equation, 7 algebraic element, 92 commutative ring, 62 digraph, 216 algebraic multiplicity, 89 compact set, 61 dimension, 3 algebraically closed field, 63 companion matrix, 108 direct sum of matrices, 13 algebraically simple complement, 26 disjoint sets, 257 eigenvalue, 90 completion lemma, 11 disjoint union, 188 angle, 35 complex number, 10 distance formula, 43 angle between x and y, 139 complex plane, 98 division algebra, 11 annihilator, 38 complex vector space, 162 division ring, 263 anti–self-adjoint, 156 continuity of the roots of a dot product, 139 associativity, 280 polynomial, 211 double dual, 70 augmented matrix, 15 convergent sequence, 254 doubly stochastic matrices, convex combination, 56 32 balanced bipartite graph, 55 convex function, 54 dual space, 38 basis, 3 convex hull, 56 edge, 53 big O notation, 255 convex set, 56 eigenspace, 77 bijection, 11 convoy principle, 164 eigenvector, 63 bilinear form, 161 core tensor, 226 elementary matrix, 19 binary operation, 261 coset, 100 elementary row operation, 14 bipartite graph, 52 Courant–Fischer principle, equivalence relation, 21 block diagonal matrix, 23 165 Euclidean metric space, 253 block matrix, 13 Cramer’s rule, 47 Euclidean norm, 141 cross product, 38 Euclidean space, 1 cyclic decomposition CANDECOMP-PARAFAC Euler’s function, 10 theorem, 109 decomposition, 227 even function, 30 cyclic group, 66 canonical form, 75 extreme point, 56 Carath´eodary’s theorem, 61 cyclic invariant subspace, Cauchy–Binet formula, 110 108 Fekete’s lemma, 53 Cauchy–Schwarz inequality, cyclic property, 72 Fibonacci numbers, 83 141 cyclic subspace, 108 field, 61 Cayley–Hamilton theorem, field extension, 63 85 De Moivre’s theorem, 273 finite field extension, 63 center of a group, 265 decomposable element, 221 finite-dimensional vector closed set, 173 decomposable tensor, 221 space, 8 codimension, 100 degree of field extension, 63 finitely generated group, 8 cofactor, 46 determinant of matrix, 42 finitely generated vector cokernel, 99 diagonal entries, 13 space, 5

283 284 Index

first isomorphism theorem, isomorphism, 34 neutral element, 9 101 nilpotent matrix, 102 Fredholm alternative, 146 Jordan block, 80 noncommutative division free transversal, 53 Jordan canonical form, 75 algebra, 11 free variable, 16 noncommutative ring, 262 Frobenius norm, 141 Kronecker delta function, 13 nonnegative matrices, 61 fundamental theorem of Kronecker product, 224 nonpivot column, 29 algebra, 274 Krylov subspace, 160 nontrivial invariant fundamental theorem of subspace, 94 invertible matrices, 19 Lagrange–Sylvester norm, 140 interpolation normal subgroup, 265 Gauss–Jordan algorithm, 20 polynomial, 123 normalization condition, 43 Gaussian elimination, 14 Laplace expansion, 43 normed linear space, 140 Gaussian field of rationals, lead variable, 16 numerically unstable, 145 11 leading entry, 14 Gaussian integers domain, 11 least squares system, 147 odd function, 30 general linear group, 67 least squares theorem, 152 open ball, 253 generalized singular value left coset, 265 ordered bases, 69 decomposition, 202 linear combination, 3 ordinary differential geometric multiplicity, 90 linear functional, 38 equations (ODE), 130, Gram–Schmidt (G–S) linear isomorphism, 34 132 algorithm, 153 linear operator, 34 orthogonal complement, 142 graph, 52 linear system, 16 orthogonal matrix, 150 group, 1 linearly dependent, 3 orthogonal projection, 145 linearly independent, 3 orthonormal basis, 142 H¨olderinequality, 179 lower triangular matrix, 29 orthonormal k-frame, 166 Hadamard conjecture, 154 LU decomposition, 52 orthonormal sets, 142 Hadamard determinant inequality, 150 Maclaurin expansion, 132 parallelogram identity, 152 Hadamard matrix, 154 mapping, 101 path, 170 Hermitian form, 161 Markov matrices, 56 perfect matching, 57 homogeneous linear matching, 55 permutation, 30 equation, 6 matrix, 1 permutation group, 30 homogeneous polynomial, 12 matrix addition, 1 permutation matrix, 32 homomorphism, 31 matrix function, 121 Perron–Frobenius hyperplane, 60 matrix multiplication, 1 eigenvector, 214 matrix of inner product, 140 Perron–Frobenius theorem, idempotent matrix, 199 matrix of the change of 207 index of nilpotency, 102 basis, 8 pivot, 14 induced norm by inner matrix polynomial, 87 pivot column, 14 product, 141 max-min characterizations of pivot position, 14 infinite-dimensional vector eigenvalues, 164 pivoting, 14 space, 7 maximal orthogonal set, 142 polar decomposition, 198 inner product, 37 metric, 141 positive definite operators, inner product space (IPS), metric space, 141 173 37 minimal polynomial, 91 primary decomposition invariant polynomials, 110 Minkowski’s criterion, 52 theorem, 94 invariant subspace, 91 modified Gram–Schmidt primitive matrix, 214 invertible matrices, 8 algorithm, 146 primitive unity root, 10 involution, 11 modular law for subspaces, principal minor, 47 IPS (inner product space), 30 projection, 35 139 module, 87 projection of x on U, 142 irreducible matrix, 208 Moore–Penrose generalized pseudospectrum, 98 irreducible polynomial, 63 inverse, 198 isometry, 155 multilinear operator, 34 QR algorithm, 153 Index 285

QR factorization, 144 sequence, 20 representative, 52 quaternions, 11 similar matrices, 112 similarity class, 76 tensor product of two vector rank, 14 singular value, 181 spaces, 221 rank 1 tensor, 221 singular value decomposition third isomorphism theorem, rank insensitivity lemma, 30 (SVD), 181 101 rational canonical form, 112 skew-symmetric matrix, 24, transition matrix, 227 Rayleigh quotient, 164 168 transpose matrix, 13 reduced row echelon form, 14 skew-symmetric tensor, 225 transposition, 31 reduced SVD (singular value span, 27 triangular matrix, 24 decomposition), 198 spanning set, 4 trivial combination, 3 REF (row echelon form), 6 special linear group, 67 trivial subspace, 3 reflection, 36 spectral decomposition, 158 trivial vector space, 5 representation matrix, 69 spectral norm, 141 Tucker model, 226 ring, 61 spectral radius, 207 ring of polynomial, 62 spectral theorem, 157 universal property, 222 root of unity, 10 spectrum, 75 upper diagonal matrix, 194 rotation, 35 standard basis, 8 upper triangular matrix, 29 row echelon form (REF), 6, stochastic matrix, 32 14 subgroup, 8 Vandermonde determinant, submatrix, 279 49 s-numbers, 102 subring, 68 Vandermonde matrix, 50 scalar, 1 subspace, 53 Vandermonde, scalar matrix, 24 Sylvester’s criterion of Alexander-Theophile, Schur’s unitary positivity, 180 50 triangularization, 160 Sylvester’s inequality, 50 vector, 1 second isomorphism symmetric matrix, 64 vector projection, 155 theorem, 101 symmetric product, 225 vector space, 1 second-order linear symmetric tensor, 225 vertex, 57 differential systems, symmetric tridiagonal 132 matrix, 50 walk, 216 Segre’s characteristic, 105 system of differential wedge product, 225 self-adjoint linear equations, 135 Weyl inequality, 166 transformation, 156 system of distinct Weyr characteristic, 105