Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R

Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R. Johnson , Ronald L. Smith , Michael J. Tsatsomeros Index More Information

Index

p-Newton, 159 completion of a partial matrix, 153 t-th Hadamard power, 128 completion to a P-matrix, 88 (convex) cone, 162 cone,3,39 (i,j)-path, 157 congruence of A, 168 (right) Perron eigenvector, 58 conjugate transpose, 1 (right) null space, 114 contiguous minors, 9 (strictly) codefinite pair, 172 convex, 3 (strictly) copositive of order m, 171 convex combination, 3 convex hull, 4 acute cone, 40 Copositive, 164 allow MSP,30 copositive, 10 allow RSP,30 copositive +,10 allow SP,25 copositive with respect to a cone K, 167 almost (strictly) semimonotone, 53 Copositive-plus, 164 almost C, 177 CP-rank, 11 almost C+, 177 critical exponent, 135 almost SC, 177 cycle of length, 59 almost diagonally dominant P-matrices, 88 cycle product, 120 almost principal submatrices, 109 cycle product inequalities, 120 associated quadratic form, 174 determinantal inequalities, 8 B-matrix, 58 diagonal Lyapunov solution, 91 block graph, 155 diagonal Lyapunov solutions, 95 block P-matrices, 88 diagonally dominant matrix, 2 buckle, 157 diagonally dominant of its column entries, 94 diagonally dominant of its columns, 93 Cayley transform, 57 diagonally dominant of its row entries, 94 Cholesky factorization, 10 diagonally dominant of its rows, 93 column (row) deleted submatrix, 16 diagonally symmetrizable, 127 column diagonally dominant, 12 directed graph, 59 common semipositivity vector, 59 directed graph of A, 121 comparison matrix, 58 dispersion, 75 complementary nullities, 114 doubly nonnegative matrices, 10, 135 completely positive matrix, 11 doubly positive matrices, 10

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-108-47871-7 — Matrix Positivity Charles R. Johnson , Ronald L. Smith , Michael J. Tsatsomeros Index More Information

Index 207

Drazin inverse, 161 Leading Implies All, 89 dual, 4, 39 leading principal minor, 2 dual of the IM matrices, 127 leading principal submatrix, 2 left endpoint matrix, 148 essentially triangular, 157 left Perron eigenvector, 58 Euclidean Jordan Algebras, 88 left semipositive, 15 eventually inverse M-matrix, 128 line in a matrix, 148 eventually nonnegative (positive), 7 Linear Complementarity Problem, 85 exceptional (copositive) matrix, 165 Linear Differential Inclusion, 87 extreme (copositive) matrix, 167 linear interpolation problem, 158 extreme copositive quadratic form, 165 linear preserver, 30 extreme point, 3 linear transformation of standard form, 179

Fan product, 8 M-matrix, 8, 58, 90 Fischer’s inequality, 8 matrix interval, 45 fronted positives, 23 measure of irreducibility, 114 mime, 58 generalized diagonally dominant matrix, 75 minimally SP,16 generalized ultrametric, 163 minimally semipositive, 27 generators, 3, 40 minor, 2 Gershgorin’s Theorem, 2 MMA-matrix, 58 group inverse, 39 monomial matrices, 15 monotone, 12, 15 H+-matrix, 12 Moore–Penrose, 39 H-matrix, 12, 58 Hadamard (entry-wise) product, 126 negative definite, 10 Hadamard power, 126 negative semidefinite, 10 Hadamard’s inequality, 8 Newton matrix, 159 half-space, 4 nonnegative matrices, 7 Helly’s Theorem, 4 nonnegative orthant, 7 Hermitian, 1 normal Jacobi matrix, 156 Hermitian adjoint, 1 normalized inverse M-matrix, 94 hidden Minkowski matrices, 58 nullity, 114 hyperplane, 4 obtuse cone, 40 identically signed class, 148 onto copositivity preserver, 179 index of A, 161 oscillatory, 155 initial minors, 9 interval from A to B, 147 P0-matrices, 84 into copositivity preserver, 179 P-LCP, 86 inverse M, 8 P-LCP*, 86 inverse M-matrices, 90 P-matrix, 8, 11, 55 inverse M-matrix, 58 P-matrix interval, 87 irreducible matrices, 7 P-problem, 55, 74 irreducible matrix, 59 P-tensors, 88 partial IM-matrix, 153 Jacobi matrix, 156 partial PP (SPP) matrices, 153 join of the (disjoint) graphs, 157 partial matrix, 153 path product, 120, 121 k-chordal, 154 path product equalities, 131 k-path, 157 path product equality triple, 129 Koteljanskii’s inequality, 8 path product inequalities, 120

208 Index

path product triple, 126 sign nonsingular, 59 Perron complement, 159 sign of a cycle, 59 Perron complements, 159 sign pattern, 24 Perron’s Theorem, 2 signature matrix, 53, 59 Perron–Frobenius eigenvalue, 90 signature similarity, 59 Perron–Frobenius theory, 6 signed directed graph, 59 polar (dual), 167 simplicial cone, 40 polyhedral, 3 skew-Hermitian, 1 polyhedral cone, 40, 162 Soules matrix, 59 positive definite, 9 special (symmetric ultrametric matrix), 163 positive diagonal Lyapunov solutions, 8 spectral radius, 1 positive front, 23 spectrum, 1 positive matrices, 7 strict path product, 120 positive semidefinite, 10 strictly copositive, 10, 164 positive stable, 8, 57, 90 strictly semimonotone, 48 principal minor, 2 strictly ultrametric, 137 principal pivot transform, 56 strictly ultrametric matrix, 163 principal submatrix, 2 strongly connected, 59 proper cone, 39, 162 strongly monotone, 30 proper principal submatrix, 2 submatrix, 2 purely strict path product matrix, 123, 130 symmetric IM, 154

Q-matrix, 65 Theorems of the Alternative, 5 qualitative P-matrices, 70 totally nonnegative, 9, 155 range symmetric, 39 totally positive, 9, 155 rank, 114 totally strict path product, 130 reducing block, 106 totally strict path product matrices, 122 redundantly semipositive, 27 transitive directed graph, 121 regular interval, 87 transitive matrix, 121 require RSP,30 transpose, 1 require SP,24 right endpoint matrix, 148 unipathic, 163 row diagonally dominant, 12, 57 unipathic graphs, 163 row positive, 15 unit simplex, 164 row-P-property, 88 univalent, 87 upper path product bound, 143 Schur complement, 3 self-dual cone, 40 vector of mixed sign, 180 self-polar, 167 vertices (vertex matrices), 147 seminegative, 14 von Neumann expansion, 96 seminonnegative, 14 seminonpositive, 14 semipositive, 12, 14 weight of a directed path, 158 semipositive cone, 39 weight of an edge, 158 semipositive interval, 46 semizero, 14 Z-matrices, 90 sgn, 93 Z-matrix, 7, 58