A and A∗ Similar, 229 Similar Via a Hermitian Matrix, 229 Similar Via A

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Index

A and A∗ is real, see AA¯ = AA¯ similar, 229 normal, see congruence-normal matrix, similar via a Hermitian matrix, 229 277 similar via a Hermitian positive definite positive semidefinite, 277, 311 matrix, 443, 489 similar to real matrix, 203 a posteriori bound, 384, 411 AAT = AT A a priori bound, 382, 405 and A is normal, 138 A2 = 0, see self-annihilating matrix AA∗ = BB∗ AB B = AU, 455 BA,71 A∗ A = B∗ B AB −BA, 71, 187, 273 B = UA, 155, 452, 457 AB versus BA AB, 71, 272 B −A eigenvalues, 65 0 A Jordan blocks, 184 A¯ 0 , 72, 277, 307 similarity, 189 conjugate normal or normal, 277 singular values, 455 AA∗ A∗ A , 144 Abel’s formula, 56 0 A absolute norm A∗ 0 , 158 D ei ei ≥ 1, 339 0 A definition, 332 AT 0 , 158, 276 dual norm is absolute, 332 AB −B¯ A¯ , 275, 307 dual norm representation, 332 AA¯ is monotone, 332, 339 = 0, see conjugate self-annihilating matrix Loewner–John matrix is diagonal, 492 = AA∗, 277 matrix norm that it induces, 358 = AA¯ on matrices, 470 and A is normal, 138 standardized, 339 real orthogonal quasitriangularization of A, symmetric gauge function, 464 106 adjacency matrix, 226 = I , see coninvolutory matrix adjoint =−AA∗, 277 classical, see adjugate eigenvalues, 267 Hermitian, see conjugate transpose Hermitian, 277 of a norm on matrices, 357

607

608 Index

adjugate analytic function 2-by-2 matrix, 22 of a matrix, 350 and nonzero eigenvector entries, 82 one-to-one, see univalent function, univalent columns are right eigenvectors, 82 schlicht, see analytic function, univalent definition, 22 univalent, 260 eigenvalue with geometric multiplicity one, Ando, T., xiv 82 angle eigenvectors of 2-by-2 matrix, 82 between Ax and Ay with x ⊥ y, 473 gradient of determinant, 23 between columns (rows) of a real normal Hermitian, 233 matrix, 132 Hermitian matrix, 145 between one-dimensional subspaces, 317 Jordan canonical form, 188 between real vectors, 15, 473 Newton’s identities, 198 between x and Ax, 474 nilpotent Jordan block, 190 principal, see principal angles nilpotent matrix, 190 annihilating polynomial, 129, 187, 191 normal matrix, 145 anticommutator, see Jordan product of a direct sum, 31 anticommute of a matrix with rank n − 1, 22 definition, 21, 75 polynomial in A, 125 intertwining relation, 112 positive definite, 233, 443 anticommuting family positive definite matrix, 145 diagonalizable matrices, 75 positive semidefinite, 233 finite, 75 positive semidefinite matrix, 145 antidiagonal, see counterdiagonal rank, 22, 80, 155 antihomogeneous, see conjugate homogeneous reverse product rule, 23 antilinear transformation, see sesquilinear rows are left eigenvectors, 82 transformation rth adjugate matrix approximate inverse, 351 and inverse of compound matrix, 29 approximation definition, 29 by a 2-sided rotation, 463 det(A + B), 29 by a Hermitian matrix, 323 identity related to compound matrix, 29 by a positive semidefinite matrix, 323 multiplicativity property, 29 by a rank one Hermitian matrix, 379 singular value decomposition, 155 by a rank-k matrix, 462 unitary, 93 by a scalar multiple of a unitary matrix, 463 unitary matrix, 145 by a singular matrix, 369 upper triangular, 129 in a unitarily invariant norm, 467 Alfred Horn’s inequalities, 260 least squares, 379 algebra generated by of a nonsingular matrix by a singular matrix, one matrix, 128, 197 461 two matrices, 74, 128, 197 unique, 461, 476 algebra of matrices, 74 unitary Procrustes problem, 463 algebraic multiplicity argument of a complex number definition, 51, 76 definition, 556 irreducible nonnegative matrix, 534, 535 principal value, 556 positive matrix, 526, 528 argument principle, 389 spectral radius of a positive matrix, 528 arithmetic-geometric mean inequality algebraic–geometric multiplicity inequality, 181 definition, 559 algebraic-geometric multiplicity inequality, 55, weighted, 559 59, 79 arithmetic–geometric mean inequality algebraically closed field, 561 determinant, 489 alternative, theorem of the, see Fredholm, Asano, K., 311 alternative Auerbach’s theorem, 334, 492

Index 609

augmented matrix, 12 complete principle of, 123 Autonne’s theorem, 152, 153, 263, 278 principle of, 79, 123, 134, 529 Autonne’s uniqueness theorem, 152 Birkhoff’s theorem, 253, 407, 549 Autonne, L., 159, 278, 279 Birkhoff, G., 553 Autonne–Takagi factorization Birkhoff–Konig˝ theorem, see Birkhoff’s A = UU T , 263, 278 theorem uniqueness in, 263 block Autonne-Takagi factorization bidiagonal, 36, 201, 204 A = UU T , 153 centrosymmetric, 2-by-2, 71 diagonal, 21, 30, 32, 202 back substitution, 216 diagonalizable, 115 backward diagonal, see counterdiagonal Gaussian elimination, 25 backward identity, see reversal matrix Gersgorinˇ theorem, 395 backward shift, 34 matrix norms, 370 balanced vector strictly diagonally dominant, 395 and singular circulant matrices, 100 triangular, 31 definition, 445 pth root, 130 null space of correlation matrix, 445 tridiagonal, 36 ball upper triangular, 202, 204 in a normed linear space, 565 block matrix of radius r, 335 positive (semi)definite, 495 unit, see unit ball criterion, 496 Ballantine, C. S., 492 Bochner’s theorem, 427 Barrett, W., xiv, xviii bordered matrix basis characteristic polynomial, 56 and norms, 321 companion matrix singular values, 197 change of, matrix, 40 definition, 26 coordinates of a vector, 39 determinant, 26, 475 definition, 3 eigenvalue interlacing, 242, 258, 444 dual, 93 eigenvalues, 56, 107, 242, 258 extend to a, 4 normal, 144 orthonormal, 16 rank, 258, 259 reciprocal, 93 boundary, 336 representation, 39 boundary value problem, 427 standard, 4 bounded set, 336, 565 Bauer, F. L., 413 Bourgeois, G., 130 Bauer-Fike theorem, 405 Braciˇ c,ˇ J., 200 Beasley, L., xii Brauer set, 415 Belitskii, G., xiv, 207, 210, 215 Brauer’s theorem, 51, 122, 186, 415, 416 Benedetti, R., 278 Brauer, A., 424 Berenstein–Veinstein inequality, Brualdi’s theorem, 419, 421 475 Brualdi, R., 175, 191, 396, 424 Bergstrom’s¨ inequality, 475 Buchheim, A., 131 Bhatia, R., 260, 564 Burke, J. V., 413 bidiagonal matrix Burnside’s theorem on matrix algebras, 74 block bidiagonal, 36 definition, 36 Cain, B., xii distinct singular values, 259 cancellation theorem lower and upper, 36 congruence, 295 unreduced, 259 ∗congruence, 291 bilinear form, 226 consimilarity, 305 biorthogonality similarity, 188

610 Index

canonical angle, see ∗congruence canonical real vectors, 559 form Cayley–Hamilton theorem ∗congruence diagonalizable matrix, 293 2-by-2 case, 130 canonical blocks, see canonical form Buchheim’s proof, 131 canonical form Cayley’s comments on the proof, 131 congruence, 294 commutative ring, 124 ∗congruence, 290 Frobenius gave first general proof, 131 consimilarity, 305 theorem, 109 definition, 41 two-variable version, 128 Jordan, 167 centralizer real Jordan, 202 definition, 213 real normal, 136 dimension, 213 real orthogonal, 137 centrohermitian, 36 real skew-symmetricc, 136 centrosymmetric real symmetric, 136 2-by-2 block, see block centrosymmetric, squared-normal matrix, 147 2-by-2 test for equivalence, 164 block structure of, 36 unitary Weyr, 211 definition, 36 Weyr, 206 eigenvalues, 199 canonical pairs similar to block diagonal matrix, 74 Hermitian–Hermitian, 295, 567 characteristic equation, 49 symmetric–skew symmetric, 295, 567 characteristic polynomial canonical ray, see ∗congruence canonical form bordered matrix, 56 Caratheodory´ theorem on convex sets in Rn, 557 coefficients, 53, 54 Caratheodory’s´ theorem, 552 definition, 49 Caratheodory–Steinitz´ theorem, see derivative Caratheodory´ theorem and eigenvalue multiplicities, 54 cardinality, 3 derivative of, 29 Carlen, E., 75 generalized, 127, 297 Carlson, D., xii positive (semi)definite matrix, 438 Carmichael and Mason’s bound, 366, 404 traces of compound matrices, 108 Carroll, Lewis, see Dodgson’s identity Choi, M. D., 75 Cartesian product of vector spaces, 370 Cholesky factorization Cassini ovals, see ovals of Cassini 2-by-2 matrix, 442 Cauchy implies QR factorization, 456 bound on zeroes of a polynomial, 365, 404 positive semidefinite matrix, 90, 441, 442 expansion of determinant, 26 Choudhury, D., xii interlacing theorem, 242, 298 Chowdhury, R., xii matrix, 38 circulant matrix determinant, 38 basic circulant permutation, 33 rank-one perturbation, 26, 66 definition, 33 separation theorem, 242 eigenvalues, 100 sequence, 328 nonsingular if diagonally dominant in any spectral theorem, 234 one row, 396 Cauchy, A., 234 nonsingularity and balanced vectors, 100 Cauchy–Binet formula, 28 properties, 33 Cauchy–Schwarz inequality spectral norm, 365, 594 case of equality, 315 spectrum of perturbation, 57 definition, 15 unitary diagonalization, 100 for a pre-norm, 330 classical adjoint, see adjugate proof for a semi-inner product, 316 Clausing, A., 476 proof for an inner product, 315 closed set, 336, 565

Index 611

closure, 336 simultaneous triangularization, 103 coefficient matrix, 12 simultaneously block triangularizable, 116 cofactor simultaneously triangularizable, 115 definition, 17 Weyr canonical form, 211 size r,17 compact column definition, 336 operations, 9 unit ball in finite dimension, 336 rank, 12 compact set, 565 column inclusion property companion matrix definition, 216, 432 alternative forms, 197 LU factorization, 216 common eigenvector, 198 positive (semi)definite matrix, 432 condition number, 386 positive definite Hermitian part, 434 definition, 194 positive semidefinite Hermitian part, 433 eigenvectors, 198 real and imaginary parts of a positive inverse, 199 semidefinite matrix, 503 irreducible, 404 sufficient condition, 436 Newton’s identities, 198 column space, 8 nonderogatory, 195 column stochastic matrix, see stochastic matrix, normal, 197 column polar decomposition, 457 combinatorially symmetric matrix, see singular values, 197 nonnegative matrix, combinatorially unitary, 197 symmetric compatible norms commutative ring, 124 characterization, 374 commutator definition, 344, 373 [A, B] = cI is impossible if c = 0, 130 matrix norm → vector norm, 347 A and A∗, 144 necessary condition, 375 commutes with, 127, 130, 190, 198, 200 spectral dominance, 347 definition, 126 vector norm → matrix norm, 344 Laffey’s theorem, 126 complementary nullities, 19, 87, 222 multiplicative, 146 complete normed linear space, 329 nearly normal matrix, 144 complete principle of biorthogonality, of a pair of Hermitian matrices, 446 see biorthogonality, complete quantum system, 447 principle of rank 1 ⇒ nilpotent, 126 complete system of invariants, 41 rank 1 ⇒ simultaneously triangularizable, completeness property, 328 126 complex number commute addition, 555 and invariant subspace, 190 argument, 556 anticommute, see anticommute conjugate, 555 definition, 21 definition, 555 quasicommute, 127 division, 555 with every unitary matrix, 130 imaginary part, 555 commuting family modulus, 555 common eigenvector, 63 multiplication, 555 definition, 62 polar coordinates, 556 normal, 135, 143 real part, 555 real triangle inequality, see triangle simultaneous real orthogonal inequality quasitriangularization, 105 unit disc, 556 real normal, 137 complex orthogonal equivalence real symmetric, 145 to transpose, 279

612 Index

complex orthogonal matrix distinct coneigenvalues, 303 2-by-2 block matrix, 93 simultaneous, 307 definition, 92 condition number for eigenvalues, 406, 411 Jordan canonical form, 276 condition number for inversion not necessarily diagonalizable, 276 companion matrix, 386 similar to a, 276 definition, 382 upper triangular ⇔ diagonal, 93 equivalence, 385 complex representation geometric characterization, 386 quaterion type, 275 normal matrix, 385 complex type, 71 positive definite matrix, 444 compound matrix conditionally positive semidefinite matrix and the characteristic polynomial, 108 definition, 484 definition, 21 Hadamard exponential, 484 det(A + B), 29 coneigenpair determinant of, 22 definition, 301 eigenvalues, 107 equivalent to real eigenpair problem, 309

of C2(A), 158 coneigenspace homogeneous of order r,22 and null space of AA¯, 304, 310 identity related to rth adjugate matrix, basis for a, 303, 309 29 definition, 302 inverse and rth adjugate matrix, 29 dimension of a, 303, 308 inverse of, 22 coneigenvalue multiplicativity property, 21 definition, 301 of a diagonal matrix, 22 real representation, 309 of a Hermitian matrix, 233 coneigenvectors of a normal matrix, 147 algorithm to compute, 309 of a positive definite (semidefinite) matrix, and eigenvectors of AA¯, 302 233 associated with nonnegative coneigenvalue, of a triangular matrix, 22 302 of a unitary matrix, 93 definition, 301 of conjugate transpose, 22 linearly independent, 308, 309, 310 of transpose, 22 linearly independent over R, 307 rank one, 22 linearly independent over C, 307 singular values, 158 conformal partition, 17 spectral radius, 108 congruence strengthened Schur inequality, 158 cancellation theorem, 295 trace of a, 108 canonical pairs, 567 compression cosquare, 294 via determinant, 445 definition, 41 via principal minor sums, 445 equivalence relation, 281 via pth compound matrix, 445 lower triangular, 223 via trace, 437, 445 nonsingular matrices, 295 concanonical form normal, see congruence-normal matrix algorithm, 305 preserves rank, 281 theorem, 305 regular part, 295 condiagonalizable regularization, 295 definition, 286, 301 simultaneous diagonalization unitarily, 301 Hermitian–Hermitian, 287 condiagonalization Hermitian–symmetric, 286, 287 algorithm for, 310 symmetric–symmetric, 287 coninvolutory matrix, 304 via nonsingular, 287 criterion, 304 via unitary, 286

Index 613

singular part, 295 theorem, 289 to a real matrix, 299 unitary matrix, 297 to conjugate, 299 congruence-normal matrix to transpose, 299 AA¯ is Hermitian, 277 type I block cosquare, 298 AA¯ is positive semidefinite, 277 unitary characterizing properties, 143 definition, see unitary congruence definition, 143 T congruence, see congruence unitary congruence canonical form, 277 ∗congruence coninvariant subspace, 262 definition, 41 coninvolution ∗congruence criterion for unitary congruence, 311 cancellation theorem, 291 definition, 38 canonical form, see ∗congruence canonical product with “real similar”, 306 form coninvolutory matrix canonical pairs, 567 complex vs. real similarity, 70 complex orthogonal, 300 consimilar to I , 304 definition, 83, 281 definition, 38 diagonalizable, 436, 457, 514 diagonalizable, 191 algorithm, 293 intertwining relation, 308 equivalence relation, 281 Jordan canonical form, 191 invariants, 292 singular value decomposition, 278 normal matrices, 296, 457 singular values, 159 preserves rank, 281 unitary congruence canonical form, 311 regularization, 292 conjugate simultaneous diagonalization homogeneous, 305 via nonsingular, 287 matrix ∗congruent to its, 299 via unitary, 286 matrix congruent to its, 299 to a real matrix, 299 conjugate linear function, 15 to conjugate, 299 conjugate of a complex number, 555 to inertia matrix, 282 conjugate partition, 172 to transpose, 299 conjugate self-annihilating matrix type I block cosquare, 297 definition, 157 unitary matrices, 457 singular value decomposition, 157 congruence canonical form conjugate transpose algorithm, 295 definition, 6 theorem, 294 conjugate-normal matrix ∗congruence canonical form block matrix criterion, 277 ∗cosquare, 290 block triangular, 268 algorithm, 292 block triangular is block diagonal, 268 canonical angle, 290 definition, 268 canonical blocks, 290 direct sum of real orthogonals, 269 canonical ray, 290 is congruence normal, 277 definition, 290 triangular is diagonal, 268 diagonalizable, 293 unitary congruence canonical form, 268, 274 generalization of Sylvester’s inertia theorem, unitary congruence invariant, 268 297 unitary congruence of two, 274 Hermitian matrix, 297 conjunctive, 281 normal matrix, 296 consimilarity positive definite Hermitian part, 436 and a real representation, 72, 310 positive semidefinite Hermitian part, 436 cancellation theorem, 305 regular part, 290 canonical form, 305 singular part, 290 complex orthogonal, 300

614 Index

consimilarity (cont.) diagonal of inverse, 444 definition, 41, 300 eigenvalue bounds, 445 real, 300 nullspace vectors are balanced, 445 to a Hermitian matrix, 306 tridiagonal to a real matrix, 306 eigenvalues, 445 to conjugate, 306 cosquare to transpose, 306 Jordan canonical form, 294 unitary, 300 type I block, 298 consistent linear system, see linear system, ∗cosquare consistent A−1 similar to A∗,86 consistent norms, see compatible norms definition, 290 continuity Jordan canonical form, 290 definition, 314 type I block, 297 eigenvalues of a matrix, 122, 564 counterdiagonal with respect to a norm, 314 definition, 33 zeroes of a polynomial, 563 coupled equations, 176, 285 continuous function Courant, Richard, 239 definition, 563, 565 Courant–Fischer theorem, 236 uniformly, 565 covariance contraction matrix, 280, 426 definition, 161, 493 of X and Y in the state R dilation to a unitary, 162 Cauchy–Schwarz inequality, 447 strict definition, 446 definition, 493 semi-inner product contriangularizable definition, 446 block, 301 Cragnolini, P., 278 definition, 301 Cramer’s rule, 24 unitarily, 301 cramped unitary matrix, see unitary matrix, convergence with respect to a norm, 325 cramped convergent matrix critical point, 425 characterization, 181 cross diagonal, see counterdiagonal criterion, 348 CS decomposition definition, 180, 348 general form, 162 convex theorem, 160 combination, 557 cycle cone, 557 definition, 400 function, 558, 559 simple directed, 400 hull, 557 trivial, 400, 418 set, 557 cyclic of index k, 537 generators, 557 unit ball of norm, 337 defect from normality, 102, 144, 147, 156 convex function, 426 defective, 77 convex matrix functions, see Loewner partial definite Hermitian matrix, see positive order, convex matrix functions semidefinite or positive definite matrix correlation definitions Wigner–Yanase, see Wigner–Yanase how to find, xiii correlation deflation correlation matrix and the power method, 81 characterization as a special Gram matrix, if one eigenvalue is known, 82 445 proof of Schur triangularization theorem, 101 definition, 434 real matrix, 74

Index 615

degree subdiagonal, 5 of a polynomial, 46 superdiagonal, 5 deleted absolute column sums, 389 diagonal entries deleted absolute row sum, 388 eigenvalues, see Mirsky’s theorem dense, 363 equal, 95 density matrix diagonal equivalence definition, 446 definition, 41 mixed state, 446 to a matrix with distinct eigenvalues, pure state, 446 75 square root is unit vector in Frobenius norm, diagonal matrix 446 commute with a, 30 DePrima, C. R., 94 definition, 30 derogatory left multiplication by a, 30 definition, 77 right multiplication by a, 30 determinant diagonalizable arithmetic–geometric mean inequality, 489 ∗congruence, 436 Cauchy expansion of, 26 all eigenvalues = 0, 58 cofactor, 17 all eigenvalues = 1, 58 definition, 8 almost, 117 derivative of, 29 ∗congruence, see ∗congruence, det(A + B), 29, 108 diagonalizable det(A + B) ≥ det A + det B, 490, 511 consimilarity, 304 Dodgson’s identity, 29 criterion, 174, 193, 194 empty matrix, 24, 507 definition, 59 Kronecker theorem, 27 part, 189 Laplace expansion of, 8, 28 rank one, 173 log det is strictly concave on positive real matrix, 203 definites, 488 real orthogonal equivalence, 155 minor, 17 real orthogonally, 94 multiplicativity, 11 simultaneously, 61 negative definite matrix, 431 unitarily, 94 positive (semi)definite matrix, 430 unitary equivalence, 150 product of eigenvalues, 51 upper triangular, 145 rank-one perturbation, 26 via real orthogonal equivalence, 155 similarity invariant, 59 diagonally dominant superadditive on positive definites, 490, circulant matrix (any row), 100, 396 511 criterion, 397 Sylvester’s identity, 27 definition, 392 upper bound, 107, 363, 394 preserved under Gaussian elimination, Hadamard inequality, 93 394 upper bound used in Fredholm integral strictly, see strictly diagonally dominant equations, 514 difference equation, 581 Wronskian, 56 differential operator Deutsch, E., 492 elliptic, see elliptic differential operator dexter-diagonal, see counterdiagonal hyperbolic, see hyperbolic differential diag, 5, 30, 479 operator diagonal linear second order, 226 backward, see counterdiagonal dilation entries, permutation, 32 to a unitary matrix, 162 main, 5 dimension, 4 secondary, see counterdiagonal Ding, C. X., 515

616 Index

direct sum of | · | 1, 343, 360 adjugate, 31 of spectral norm, 362 matrix pairs, 568 dual norm of matrices, 30 deﬁnition, 329 of subspaces, 2 duality theorem, 338 directed path, 400 is a norm, 330 discriminant of absolute norm, 332 determinant of moment matrix, 128 of Euclidean norm, 331 quadratic polynomial, 50, 75, 264 of k-norm on vectors, 334

distance of l p norm, 331 to nearest Hermitian matrix, 476 of max norm, 331 to nearest rank-k matrix, 462 of sum norm, 331 to nearest scalar multiple of a unitary matrix, self-dual norm, 331 463 duality theorem to nearest singular matrix, 369, 461 application, 339, 343, 344, 353, 355, 359, to nearest unitary matrix, 476 361, 364 Ðjokovic,´ D., 100 theorem, 338 Dodgson’s identity, 29 Dungey, J. W., 130 Domanov, I., xiv, 191 dominant eigenvalue, 532 e, see vector, all ones dot diagram, 172, 206 edge, 226 dot product, see inner product eigenpair, 44 doubly stochastic matrix eigenspace, 76 Birkhoff’s theorem, 253 eigenstate compact convex set, 548 definition, 447 convex combination of permutation matrices, eigenvalue 549, 552 semisimple, 181 definition, 253, 548 eigenvalue inclusion sets, 393 not unistochastic, 413 eigenvalues permutation matrices are extreme points, 2-by-2 matrix, 50, 128 552 A + B, 239 reducible, 552 AB vs. BA,65 semigroup, 551 adj A, 145 spectral matrix, 552 and dominating nonnegative matrix, 526 unistochastic, 257, 407 and singular values, 347 doubly substochastic matrix bordered matrix, 56 definition, 550 bounds on diagonal entries, 238 dominated by doubly stochastic matrix, 550 continuity, 122, 564 Drazin inverse definition, 44 characterized by 3 identities, 186 derivative, 409 definition, 185 diagonal entries, see Mirsky’s theorem limit, 189 distinct, 51, 57, 60, 75, 124 polynomial in A, 186 moment matrix criterion, 129 projections, 189 number of, 129 reciprocal eigenvalues, 186 sign pattern requires, 129, 130 semiconvergent matrix, 191 inclusion set, 388, 396, 399, 412, 413 Drazin, M. P., 130 location near n−1 tr A, 146 dual basis, 93 low-rank matrix, 65 dual matrix norm multiplicity, 51 definition, 358 zero derivatives of characteristic norm induced by absolute norm, 361 polynomial, 54

of l1 matrix norm, 359 not in interior of any Gersgorinˇ disc, 397

Index 617

on boundary of Gersgorinˇ set, 397, 398 triangular, see triangular equivalence perturbation of, 405 unit triangular, see unit triangular equivalence polynomial in A, 108 unitary positive (semi)definite matrix, 430 definition, see unitary equivalence product is determinant, 51 equivalence class, 40 rank-one perturbation, 52, 122, 186 equivalence relation real matrix, 145 definition, 40 real matrix, odd size, 55 reflexive, 40 reversal matrix, 145 symmetric, 40 secondary, 527 transitive, 40 semisimple, 76, 174, 368, 552 equivalent norms, 327 simple, 76, 408, 409 essentially nonzero derivative of characteristic Hermitian, 7, 89, 233 polynomial, 54 nonnegative matrix sum is trace, 50 definition, 532 tridiagonal matrix, zero main diagonal, 81 dominant eigenvalue, 532 Weyl product inequalities, 370, 454 Gram matrix, 539 eigenvector triangular, 32 2-by-2 matrix, 128 Euclidean algorithm, 130, 192 common, 63, 120 Euclidean length or norm, see norm, companion matrix, 198 Euclidean definition, 44 Euler’s theorem, 143 left, 77 exchange matrix, see reversal matrix normalized, 45 extreme point, 552, 557 unit, 45 elementary row/column operations, 9 family of matrices elementary symmetric function definition, 62 and Hadamard’s inequality, 514 irreducible, 62 definition, 54 reducible, 62 ellipsoid Fan Loewner ellipsoid of a norm, 491 determinant inequality, 512 positive definite matrix, 491 dominance theorem, 467 elliptic differential operator eigenvalue majorization inequalities, 250, definition, 296 259, 260 local maxima/minima, 480 k-norms, 466 positive definite discretization, 429 normalizable matrices, 300 Elsner, L., 148, 223, 564 theorem on location of eigenvalues, Elton, J. H., 300 526 empty matrix Fan, K., 300, 476 determinant, 24, 507 Faßbender, H., 278 EP matrix Fearnley-Sander, D., 320 definition, 158 Fejer’s´ uniqueness theorem, 480 normal matrix is EP, 146 Fejer,´ L., 485 rank-principal, 296 Ferrers diagram, see dot diagram rank-principal matrix, 158 field equal diagonal entries, 95 algebraically closed, 561 equilibrated, 336 definition, 1 equivalence field of values and rank, 13, 158, 189 definition, 372 definition, 41 Fill, J., xiv diagonal equivalence, see diagonal finite-dimensional quantum system, see equivalence quantum system

618 Index

finitely generated convex set, 557 Gaussian matrix Fischer inequality definition, 443 block generalization, 507 determinant, 444 proof via Cholesky decomposition, 514 positive definite, 444 proof via Minkowski inequality, 513 gcd matrix reverse, 514 definition, 483 theorem, 506 positive semidefinite, 483 Fischer, E., 239 Gelfand formula FitzGerald, C. H., 505 matrix norm, 349 Fletcher, R., 175 pre-norm, 372 Fonseca, C. M., xiv vector norm on matrices, 372 forward shift, 34 general linear group, 14 forward substitution, 216 generalized coordinates, 285 Fourier matrix, 100 generalized inverse, 185, 453 Fredholm generalized matrix function, 9 alternative, 16 generalized matrix norm, see vector norm on Friedland, S., 75 matrices Frobenius generalized permutation matrix inner product, 321 definition, 33

norm isometry for l p norm, 334 absolute norm, 342 subgroup of GL(n, C), 33 deﬁnition, 341 uniqueness of factors, 221

l2 matrix norm, 321 generators of a convex set, 557

AB2 = BA2 sometimes, 370 geometric mean tr A∗ A, 7, 133 definition, 445 unitarily invariant, 342 G(A, A−T ) is complex orthogonal and rank inequality, 13 coninvolutory, 445 theorem, 146 G(A, A¯) is real, 445 Frobenius normal form, 532 G(A, B) = G(B, A), 445 Fuglede–Putnam theorem unique, 445 alternative proof, 143 geometric multiplicity AX = XB and X is normal, 148 and Gersgorinˇ discs, 396 theorem, 138 definition, 76 Fujii, M., 370 positive matrix, 525 full-rank factorization geometric–algebraic multiplicity inequality, definition, 13 181 Hermitian matrix, 234 geometric-algebraic multiplicity inequality, 55, normal matrix, 145 59, 79 singular value decomposition, 450 Gerasimova, T. G., xiv, 279 thin singular value decomposition, 456 Gersgorin,ˇ S., 396 function Gersgorinˇ concave, 558 circles, 389 convex, 558 discs strictly convave, 558 and geometric multiplicity, strictly convex, 558 396 fundamental theorem of algebra, 46, 561 and rank, 392, 394 definition, 388 Garcia, S. R., 100, 273 disjoint, 388, 390 gauge function, 335 set, 388, 389 Gaussian elimination theorem definition, 25 block form, 395 preserves strict diagonal dominance, 394 statement and proof, 388

Index 619

version if geometric multiplicity > 1, 396 block matrix version, 513 weighted, 390 case of equality, 93, 443, 505, 506 Gerstenhaber’s theorem, 128 column norm version, 506 Givens rotation, see plane rotation diagonal product version, 505 Givens’ method, 99 proof via arithmetic–geometric mean Goldberg, M., 381 inequality, 505 Goodson, G., xvi, 148, 191 proof via Cholesky decomposition, 443 Google matrix proof via QR factorization, 93 eigenvalues, 57 proof via Schur complement of bordered Jordan canonical form, 190 matrix, 514 Gram matrix Hadamard power matrix definition, 441 definition, 481 essentially nonnegative, 539 positive semidefinite, 481 inner product, 446 quadratic inequalities, 502 positive (semi)definite matrix, 440, 441 Hadamard product representation of a correlation matrix, 445 commutative, 477 Gram–Schmidt process, 93 definition, 371, 477 definition, 15 distributes over matrix addition, 477 minimal polynomial, 196 entrywise absolute value of a matrix, 482 graph entrywise exponential of a matrix, 481, 484 directed, 399 entrywise functions of a matrix, 481 undirected, 226 entrywise reciprocal of a matrix, 482 greatest common divisor, see gcd matrix entrywise square root of a matrix, 482 Greub–Reinboldt inequality, 473 rank inequality, 483 Grone, R., 148 trace representation of sesquilinear form, 479 group trigonometric moments, 477 bounded multiplicative group of matrices, Hadamard, J., 485 see Auerbach’s theorem Hadamard–Fischer inequality, 507 definition, 85 Hald, O. H., 260 general linear, 14 half-space, 558 Heisenberg, 174 Hall, F. J., 396 real orthogonal, 85 Halmos, P., 191 unitary, 85 Halperin, I., 75 Gruenberg, K. W., 130 Hankel matrix, 35, 427 Gruss¨ inequality, 322 Harris, L., 131 Grunsky inequalities Hartley matrix, 100 definition, 260 Hausdorff moment sequence, 427 exponentiated, 502 Hawkins, T., xiv mixed problem, 286 Haynsworth’s theorem Gu, D., 424 generalized, 437 guess and check, 309 inertia and Schur complement, 298 Guralnick, R., xiv Heisenberg group, 174 Hadamard exponential matrix uncertainty principle, 130, 447 conditionally positive semidefinite matrix, Herman, E., xiv 484 Hermite normal form, see reduced row echelon definition, 481 form positive definite/semidefinite, 481, 483 Hermite, C., 215 quadratic inequalities, 502 Hermitian Hadamard inequality adjoint, see conjugate transpose and elementary symmetric functions, 514 essentially, 7, 89 and permanent, 93 projection, 38, 124, 232, 233

620 Index

Hermitian matrix Hilbert–Schmidt norm, 341 canonical pairs, 567 Hlawka’s inequality, 319 characterization, 228 Holder’s¨ inequality, 414, 559 commuting family, 229 Hoffman, A. J., 413, 476 ∗congruence canonical form, 297 Hoffman–Wielandt theorem definition, 7 eigenvalues, 407 eigenvalues are real, 135 unitarily invariant norms, 468 essentially, 233 singular values, 451 every nonzero principal minor of size r has Hong, Y. P., xii, 108, 279, 300, 311, the same sign, 234 492 full-rank factorization, 234 Horn, Alfred orthonormal eigenvectors, 234 inequalities, 260 positive semidefinite part, see positive Horn, R. A., 108, 148, 278, 279, 300, 311, 457, semidefinite part of a Hermitian matrix, 492, 505, 569 323 Householder matrix product of two, 229, 489 definition, 87 product with a positive definite, 446, eigenvalues, 88 486 factorization of an arbitrary matrix, 93 product with a positive semidefinite, 486 Householder transformation product with symmetric, 306 definition, 95 rank-principal, 20, 234 Hsu, P. L., 311 signs of principal minors of size rank A, 234 Hua, L.-K., 278 similar to a, 489 Huang, L., 311 spectral theorem, 135, 229 Huang, Z., 75 uniqueness of diagonalization, 134 hyperbolic differential operator unitarily diagonalizable, 135 definition, 296 Hermitian part hyperplane, 558 definition, 7, 227 positive (semi)definite, 433 idempotent, see projection ∗congruence canonical form, 436 identity LU factorization, 437 matrix, 6 row and column inclusion properties, 433, Ikramov, Kh., xiv, 148, 278 434 ill conditioned positive definite, 504 matrix inversion, 382 ∗congruence canonical form, 437 imaginary axis, 556 H(A) positive definite ⇔ H(A−1) positive imaginary part, 7, 48 definite, 436 imaginary part of a complex number, 555 Toeplitz decomposition, 227, 289, 567 inclusion principle, 246 unitarily invariant norm bound, 476 inclusion set Hessenberg matrix singular values, see singular values, inclusion diagonalizable, 174 set unitarily similar to a, 96 inconsistent linear system, see linear system, unreduced, 35, 82, 174 inconsistent upper or lower, 35 indecomposable Hessian, 225, 558 permutation similarity, 182 Hessian matrix, 426 indefinite Hilbert matrix positive and negative eigenvalues, 234 condition number, 386 simultaneous diagonalization, 299 definition, 38 indefinite Hermitian matrix determinant, 38 definition, 430 inverse, 38 complex, 230 positive definite, 483, 484 indefinite real symmetric matrix positive semidefinite, 483 definition, 231

Index 621

index interlacing theorem how to use the, xiii Cauchy, 242, 298 index of an eigenvalue inverse, 243, 245 block upper triangular matrix, 183 singular values, see singular values, deﬁnition, 169 interlacing equivalent deﬁnitions, 183 Weyl, 241 exponents in minimal polynomial, 200 intertwining relation

rq (A,λ) = rq+1(A,λ), 175 and polynomials, 111 Segre characteristic, 170 AX∗ = X ∗ B and X ∗ A = BX∗, 148 index of nilpotence D = AB − BAT and AD = DAT , at most n, 109 189 definition, 38 definition, 112 index of primitivity, see primitive matrix, index Fuglede–Putnam theorem, 138 of primitivity linear matrix equation, 111 index set, 16 normal matrices and AX = X B¯ , 142 indicator matrix, 399 uniqueness in Schur triangularization, indicial equation, 581 113 induced matrix norm invariant bounds between two, 352 complete system, 41 by an absolute norm, 358 definition, 41 definition, 344 subspace, 62, 190, 262 distance to a nearest singular matrix, 369 invariant factors, 200 equal norms, 354 inverse is minimal, 356 2-by-2 matrix, 22 spectral norm is the only self-adjoint, 358 approximate, 351 spectral norm is the only unitarily invariant definition, 14 norm, 357 errors in computing, 382 uniqueness of inducing norm, 356 left, 14 unital, 343 matrix similar to its, 173 inequality partitioned matrix, 18 Cauchy–Schwarz, 15 polynomial in A, 110 inertia rank one adjustment, 19 and canonical angles, 297 reverse monotone on positive definite Haynsworth’s theorem, 298 matrices, 495, 504, 505 Hermitian matrix, 281 right, 14 Sylvester’s law of, 282 inverse eigenvalue problem inertia matrix bordered Hermitian matrix, 243 definition, 282 Jacobi matrix, 260 inner product rank one perturbation of a Hermitian matrix, axioms, 315 245 definition, 15, 315 trace = eigenvalue sum (Mirsky’s theorem), Euclidean, 15 68 Gram matrix, 446 inverse interlacing, 243, 245 properties, 315 invertible matrix, see nonsingular semi-inner product, see semi-inner product matrix sesquilinear, 315 involution standard, 15 A =−KAK, 199 usual, 15 A = KAK, 199 inner product space definition, 38 definition, 15, 316 diagonalizable, 199 interior point symmetric, 274 definition, 336 involutory, 38 unit ball, 336 Ipsen, I., xiv

622 Index

irreducible Jordan decomposition definition, 402 diagonalizable part, 189 family, 62 existence, 182 nonnegative matrix, see nonnegative matrix, nilpotent part, 189 irreducible uniqueness, 189 irreducible normal form, 532 Jordan matrix, 164 irreducibly diagonally dominant Jordan product definition, 403 anticommuting family, 75 isometry definition, 75, 446 condition number = 1, 386 of positive definites need not be positive Euclidean, 85 definite, 446 for a norm, 334 quantum system, 447 determinant has modulus one, 334, 491 Jordan, C., 175 eigenvalues have modulus one, 334 Jordan, P., 320 similar to a unitary matrix, 334, 491, 492

for an l p-norm, 334 k-norm group, 334 vector norm isomorphism, 4 deﬁnition, 320 isotropic vector, 7, 148, 274 dual, 340 dual norm, 334

Jn, see matrix, all ones permutation invariant and absolute, Jacobi 321 identity, 24 Kakeya’s theorem, 366 method, 98 Kantorovich inequality Jacobi matrix equivalent to Wielandt inequality, 472 deﬁnition, 36 proof, 471 distinct eigenvalues, 174 scalar version, 473 inverse eigenvalue problem, 260 statement, 470 Jacobson’s lemma, 127, 130, 189, 200 Kaplansky, I., 279 Jacobson, N., 130, 278 Kellogg, R. B., 533 Jensen’s inequality, 333, 560 kinetic energy, 285 Jespersen, D. C., xiv Kirk, W. A., 324 Johnson, C. R., 94, 100, 148, 191, 381, 429, Kittaneh, Fuad, 200 492, 505, 533 Kojima’s bound, 367, 404 Jordan block Kolotilina’s theorem, 423 deﬁnition, 164 Kolotilina, L. Yu., 424 inverse, 173 Konig,˝ D., 553 unitarily similar to a symmetric matrix, 271, Kosaki, H., xiv 272 Koteljanski˘ı inequality, 507 Jordan canonical form Krass, D., xii AB versus BA, 184 Krause, G., 564 complex orthogonal matrix, 276 Krein–Milman theorem, 557 of a direct sum, 182 Krishnaiah, P. R., 477 of a matrix, 167 Kronecker optimality, 183 theorem for bordered determinants, 27 rank-one perturbation, 186 Krylov sequence, 221 real, 202 Kubo, F., 370 real orthogonal matrix, 276 Kuzma, B., 200 self-annihilating matrix, 172 Ky Fan k-norms, see Fan, k-norms skew-symmetric matrix, 276

symmetric version, 271 l p-norm, 320 theorem, 167 Laffey’s theorem, 126, 127

Index 623

Laffey, Thomas J., 130 errors in solution of a, 383 Lagrange inconsistent, 12 interpolation formula, 38 least squares solution, 462 interpolation polynomials, 38 sensitivity of solution, 385 Lagrangian, 44 solution, 12 Laguerre, E. N., 319 linear transformation Lancaster, P., 569 definition, 5 Lanczos tridiagonalization, 221 matrix represents a, 5 Laplace linearly dependent expansion, 8 definition, 3 expansion theorem, 28 list of vectors, 3 LDU factorization set of vectors, 3 proof of Sylvester’s criterion, 442 linearly independent theorem, 218 definition, 3 leading entry, see reduced row echelon list of vectors, 3 form set of vectors, 3 leading principal minor, see principal minor, Lippert, R. A., xiv, 191, 279 leading Liping, H., see Huang, L. leading principal submatrix, see principal list of vectors submatrix, leading length, 3 least common multiple, 196 linearly dependent, 3 least squares approximation, 379 linearly independent, 3 least squares solution of a linear system, 453, span, 3 462 sublist, 3 left eigenvector Littlewood, D. E., 211, 215 definition, 77 Loewner ellipsoid if it is also a right eigenvector, 77 associated with a norm need not be a right eigenvector, 77 definition, 491 left Perron vector, see Perron vector, left Loewner partial order length and contractions, 494 Euclidean, 15 concavity of Schur complement, 504 of a directed path, 400 convex matrix functions, 505 of a vector, see norm convexity of inverse, 502 of a word, 97 convexity of inverse square root, 505 Levy–Desplanques theorem, 352, 392 convexity of negative square root, 505 Li, C. K., xiv, 75, 260 convexity of square, 501 Li, L., 424 criterion, 494 Li, Z., xiv, 131 definition, 493 Lidskii’s eigenvalue majorization inequalities, H −1[α] ! (H[α])−1, 499 250, 259, 260 monotone matrix functions and difference Lidskii, B. V., 260 quotients, 505 Lieb, E., 75 monotonicity of determinant, 495 Lim, T. C., xiv monotonicity of Schur complement, 504 limit point, 336 monotonicity of square root, 495, 501, linear combination 505 definition, 2 monotonicity of trace, 495 nontrivial, 2 reverse monotonicity of inverse, 495, 504, trivial, 2 505 linear equations, see linear system Loewner, C., 505 linear space, see vector space Loewner–John matrix linear system absolute norm, 492 consistent, 12 applications, 492

624 Index

Loewner–John matrix (cont.) Marsli, R., 396 associated with a norm Mastnak, M., 101, 130 deﬁnition, 491 Mathias, R., xii, xiv, 260

l p norms, 492 matrix symmetric gauge function, 492 adjacency, 226 Lomonosov, V., 75 all ones, 8, 435 loop, see cycle, trivial eigenvalues, 65 Lorentz transformation, 300 augmented, 12 lower triangular congruence, 223 backward shift, 34 lower triangular matrix bidiagonal, see bidiagonal matrix commutes with, 31 block, 18 definition, 31 block centrosymmetric, 2-by-2, 71 LU factorization, 216 block diagonal, 21, 30, 32 not square, 31 block triangular, 31 unit, 31, 218 bordered, 26, 107 LPU factorization, 219 Cauchy, 38 Lowner,¨ K., see Loewner, C. centrohermitian, 36 LPDU factorization, 221 centrosymmetric, 36 LPU factorization, 219, 222 change of basis, 40 LQ factorization, 90 circulant, see circulant matrix LU factorization coefficient, 12 definition, 216 companion, 194 min matrix, 442 complex orthogonal, 92 nonexistence, 217 complex type, 71 PLU factorization, 219 compound, 21 positive (semi)definite Hermitian part, 437 conditionally positive semidefinite, see tridiagonal matrix, 222 conditionally positive semidefinite uniqueness, 218 matrix lub norm, see induced matrix norm ∗congruence canonical blocks, 289 LU P factorization, 219 congruence-normal, see congruence-normal matrix M-matrix coninvolutory, 38 definition, 533 conjugate self-annihilating, see conjugate equivalent characterizations, 533 self-annihilating matrix nonnegative inverse, 533 conjugate-normal, see conjugate-normal magic square, 532 matrix main diagonal, 5 consimilarity canonical blocks, 305 majorization convergent, see congergent matrix definition, 248 correlation, see correlation matrix diagonal entries and eigenvalues, 249, 251 covariance, 426 diagonal entries and eigenvalues (block definition, 5 version), 259 diagonal, 30 eigenvalues of Hermitian part, 254 diagonalizable, 59 geometric characterization, 254 direct sum, 30 multiplicative, 454 doubly stochastic, see doubly stochastic Marcoux, M., 101, 130 matrix, see doubly stochastic matrix Marcus-Thompson theorem, 146 essentially nonnegative, see essentially Markovian matrix nonnegative matrix definition, 443 essentially triangular, 32 determinant, 443 forward shift, 34 inverse, 443 Fourier, see Fourier matrix positive definite, 443 Gaussian, see Gaussian matrix

Index 625

Google, see Google matrix positive semidefinite, see positive Gram, see Gram matrix semidefinite matrix Hadamard exponential, see Hadamard power-bounded, see power-bounded matrix exponential matrix principal submatrix, 17 Hadamard power, see Hadamard power product of “real similar” and coninvolutory, matrix 306 Hankel, 35, 427 product of Hermitian and symmetric, 306 Hartley, 100 product of symmetric and skew symmetric, Hermitian, 7 276 projection, 38 product of two Hermititians, 229 Hermitian part, 7 product of two skew symmetrics, 276 Hessenberg, 35 product of two symmetrics, 178, 271 unreduced, 35 projection, 38, see projection Hessian, 426 proper rotation, see proper rotation matrix Hilbert, 38 quasi-triangular, 103 Householder, 87 quasidiagonal, 32 identity, 6 quasitriangular, 32 imaginary part, 7 quaternion type, 275 inverse of partitioned, 18 rank in a partitioned, 20 invertible, see nonsingular matrix rank-principal, 20 involutory, 38 real, 51 irreducible, 402 deflation via non-real eigenvalue, 74 Jordan, 164 real orthogonal, 84, 137 Jordan block, 164 real part, 7 linear transformation, of a, 5 real skew-symmetric, 136 lower triangular, 31 real symmetric, 136 Markovian, see Markovian matrix reducible, 402 nearly normal, 144 reversal, 33 negative definite, see negative definite scalar, 6, 30 matrix self-annihilating, see self-annihilating matrix negative semidefinite, see negative similarity, 58 semidefinite matrix similarity canonical blocks, 167 nilpotent, 38 singular, 14 nonderogatory, 178 skew-centrohermiitian, 36 nonnegative, see nonnegative matrix skew-centrosymmetric, 36 nonnegative diagonal, 30 skew-Hermitian, 7 nonsingular, 14 skew-Hermitian part, 7 normal, 7 skew-orthogonal, 92 normal and skew symmetric, 146 skew-perhermitian, 36 normal and symmetric, 146 skew-persymmetric, 36 orthogonal, 7, 92 skew-symmetric, 7 orthostochastic, see orthostochastic matrix, skew-symmetric and normal, 146 see orthostochastic matrix skew-symmetric part, 7 pair, 568 spectral, see spectral matrix Palais, 92 square, 5 partitioned, 17 square of nilpotent, 310 perhermitian, 36 squared-normal, see squared-normal matrix permutation, see permutation, matrix stochastic, see stochastic matrix persymmetric, 36 strictly positive, see positive matrix block triangular, 31 positive definite, see positive definite matrix lower triangular, 31 positive diagonal, 30 upper triangular, 31

626 Index

matrix (cont.) unitarily invariant, see unitarily invariant symmetric, 7 matrix norm symmetric and normal, 146 upper bound for spectral radius, 347 symmetric part, 7 max matrix Toeplitz, 34, 427 definition, 436 triangular, 31 Hadamard reciprocal is positive tridiagonal, see tridiagonal matrix semidefinite/definite, 436 tripotent, 196 max norm, see or unistochastic, see unistochastic max{a, b}, 233 matrix maximum column sum matrix norm unitary, 7, 84 definition, 344 unitary and skew-symmetric, 139 induced by sum norm, 344 unitary and symmetric, 139 maximum row sum matrix norm upper triangular, 31 definition, 345 Vandermonde, 37, 128 induced by max norm, 345 weakly irreducible, 418 McCoy zero, 6 theorem, 127, 128 zero-one, 174 little theorem, see little McCoy theorem matrix exponential proof, 119 definition, 350 McCoy, N., 130 determinant, 369 mean matrix norm definition, 319 adjoint, 357 Mellendorf, S., 396 axioms, 340 Mercer’s theorem, 478 block matrices, 370 Merino, D. I., xiv, 279, 311 bounds on zeroes of a polynomial, 365 min matrix compatible with a vector norm, 344 and gcd matrix, 483 constants between, 365 definition, 435 dual of a matrix norm, 360 positive semidefinite, 435, 436, equal to spectral radius, 368 442 Frobenius norm, 321, 341 min-max theorem, 236 generalized, see vector norm on matrices minimal matrix norm Hilbert–Schmidt norm, 341 definition, 355 induced by absolute norm, 368 is induced, 356 n induced by l p norm on C , 363 minimal polynomial induced norm, see induced matrix norm definition, 192

l1 matrix norm, 341 dimension of subalgebra, 197 l2 matrix norm, 321, 341 direct sum, 196 lub norm, see induced matrix norm Gram–Schmidt process, 196 max of several, 357, 363 rank-one matrix, 200 minimal, see minimal matrix norm minimally spectrally dominant norm monotone on the positive orthant, 368 deﬁnition, 378

N∞(·) matrix norm, 343 spectral characteristic, 378 not a convex set, 363 Minkowski operator norm, see induced matrix norm determinant inequality, 510 scalar multiple of vector norm, 373 product inequality, 511, 560 Schur norm, 341 sum inequality, 320, 560 self-adjoint, 357 minor spectral norm, see r deﬁnition, 17 trace norm, see trace norm principal, see principal minor unital, see unital matrix norm size r,17

Index 627

minors algebraic, 51, 76 quadratic relations between, 28 geometric, 76 Mirsky’s theorem of a singular value, 151 diagonal entries & eigenvalues, general, 68, of an eigenvalue, 51 75 root of a polynomial, 46 eigenvalue perturbation, 468 Mirsky, L., 75, 476 Nakayama, T., 311 mixed state, see density matrix nearly normal matrix modulus of a complex number, 555 definition, 144 moment matrix tridiagonal and Toeplitz, 144 and the discriminant, 128 negative definite matrix definition, 128 definition, 430 of order m, 129 determinant, 431 moment sequence trace, 431 Hausdorff, 427 negative semidefinite matrix Toeplitz, 427 definition, 430 moments Nehari’s theorem, 261 of eigenvalues, 109, 126 Neumaier, A., xiv of zeroes, 126 Neumann series, 365, 595 trigonometric, 260 Newton’s identities convolutions and Hadamard products, 477 and unitary equivalence, 455 momentum operator, 130 approximate version, 130 monic polynomial, 46 proof, 126 monotone matrix proof via companion matrix, 198 definition, 533 nilpotence product of two monotone matrices is index of, 38, 109 monotone, 533 nilpotent monotone matrix function, see Loewner partial adjugate, 190 order, monotone matrix functions block matrix, 128 monotone norm, see absolute norm definition, 38 definition, 332 index, 109 is absolute, 332 part, 189 weakly monotone, 340 self-annihilating, 157 monotonicity theorem, 238, 241 square, 310 Montel’s bound, 365 zero eigenvalues, 48, 107 Moore–Penrose inverse node, 226, 400 axioms, 453 nondecreasing rearrangement, 249 definition, 453 nondefective, 77 in a Schur complement, 453 nonderogatory least squares solution of a linear system, 453 commute with A and A∗, 190 limit of ordinary inverse, 454 companion matrix, 195 positive (semi)definite block matrices, 502 definition, 77, 178 Moro, J., 413 Jordan and Weyr canonical forms are Morrison, see Sherman–Morrison–Woodbury identical, 206 formula matrix commutes with, 178 Motzkin, T., 130 similar to transpose, 180 Moutard’s theorem, 480 unreduced Hessenberg matrix, 82, 174 Moutard, Th., 480, 485 Weyr characteristic, 206 multilinear function, 11 nonincreasing rearrangement, 249 multiplicative commutator, 146 nonnegative matrix multiplicity best rank-one approximation, 538

628 Index

nonnegative matrix (cont.) limits of l p-norms, 334 combinatorially symmetric, 545 L∞-norm, 321 commute with a positive matrix, 531 l∞-norm on matrices, 342 cyclic of index k, 537 L p-norm, 321 deﬁnition, 519 l p-norm irreducible, 532, 533 Loewner–John matrix, 492 derivative of spectral radius, 538 Schatten p-norms, 465

general limit theorem, 546 l∞-norm, 318, 320 main diagonal entries, 536 matrix norm, see matrix norm positive main diagonal entries, 541 monotone, see monotone norm spectrum, 535 not derived from an inner product, 318 irreducible normal form, 532 of a vector of norms, 324 magic square, 532 polyhedral, 336 Perron root, 531 Schatten p-norm, 465 Perron–Frobenius theorem, 534 standardized, see standardized norm positive eigenvector, 522, 523, 530, 531, 548 sum norm, 318, 320 power method, 545 unitarily invariant, 322 primitive, see primitive matrix usual, 15 principal submatrix, 521 weakly monotone, see weakly monotone reducible, 532 norm

spectral radius is an eigenvalue, 529 weighted l p-norm, 322 spectrum, 532 norm on matrices nonsingular matrix compatible with some vector norm, 374 deﬁnition, 14 Hadamard product norms, 371

equivalent conditions, 14 l∞-norm on matrices, 372 strict diagonal dominance, 352 monotone on the positive orthant, 368 nontrivial linear combination similarity invariant, 378 deﬁnition, 2 spectrally dominant, 374 norm normal absolute, see absolute norm real representation, 71 axioms, 314 normal eigenvector

bounds between two l p, 333 definition, 124, 143 characterization of derived norm, 318, 324 normal matrices, 144 compatible with a matrix norm, 344 singular value, 158 compatible with a non-matrix norm, 375 normal equations define via a basis, 321 and least squares, 462 definition, 314 may be poorly conditioned, 386 derived from an inner product, 316 normal matrix Hlawka’s inequality, 319 ∗congruence, 457 parallelogram identity, 317 2-by-2, 131, 143 polarization identity, 318 A∗ = p(A), 142 dual, see dual norm AA¯ is real, 138 equivalent, 327 AB vs. BA, 145 Euclidean, 15, 320 AB = 0 ⇒ BA = 0, 147 isometry for a, 334 angles between pairs of columns and pairs of isometry group of a, 334 rows, 132 k-norm, see k-norm, see k-norm block triangular, 132

l1-norm, 318, 320 bordered, 144 l p-norm, 320 characterizations, 148 L1-norm, 321 closure properties, 131 L2-norm, 321 commute with conjugate, 138 l2 matrix norm (Frobenius norm), 321 commute with transpose, 138 l2-norm, 320 commuting family, 135, 137, 143

Index 629

∗congruence canonical form, 296 Ky Fan coined term, 300 congruence normal, see congruence-normal normalized vector, see vector, normalized matrix normed linear space, 314 conjugate normal, see conjugate-normal normed vector space, see normed linear space matrix null space convex hull of eigenvalues, 145 and invariant subspace, 190 definition, 7, 131 definition, 6 diagonal entries and eigenvalues, 257 intersections, 8, 454 diagonalizable, 174 normal matrix, 146 dilation to a, 144 of a seminorm, 317 eigenvalues on diagonal, 145 orthonormal basis for the, 453 field of values, 145 null spaces full-rank factorization, 145 intersection of, 8 Hermitian, 141 nullities left and right eigenvectors, 134 law of complementary, 19, 87 nondefective, 134 nullity normal eigenvectors, 143 definition, 6 null space and range, 146 numerical radius orthonormal eigenvectors, 134, 234 power inequality, 379 polar factors commute, 456 projection, 380 polynomial in A, 142 properties, 378, 379 principal minors of size rank A lie on same unitary similarity invariant, 379 ray, 145 numerical range, see field of values principal submatrix, 144 product of two, 455 O’Meara, K. C., xii, xiv, 211, 215 QR factorization, 155 observables, 447 rank one, 144 Olesky, D., xii rank principal, 145 Olkin, I., 457 real, 136 ones canonical form, 136 matrix of all, see matrix, all ones commuting family, 137 vector of all, see vector, all ones orthogonal, 137 open set, 336, 565 skew-symmetric, 136 operator norm, see induced matrix norm symmetric, 136, 143 Oppenheim, A., 515 similar, 142 Oppenheim inequality singular value decomposition, 156 case of equality, 515 skew-Hermitian, 141 inequality, 509 skew-symmetric, 146 Oppenheim–Schur inequality spectral theorem, 133 case of equality, 515 symmetric, 146 inequality, 509 triangular is diagonal, 132 ordinary differential equations, 176, 199 tridiagonal, 147 orthogonal Toeplitz, 144 complement, 16 uniqueness of diagonalization, 134 diagonalization, 272 unitarily diagonalizable, 133, 174 group, 92 unitarily similar, 142 list, 83 unitary, 141 list of vectors, 15 Wigner–Yanase skew information is matrix, 7, 92 nonnegative, 448 projection, see Hermitian projection zero-one matrix, 148 unitary, 140 normalizable matrix vectors, 15 definition, 300 orthogonal matrix, see complex orthogonal determinant inequalities, 514 matrix or real orthogonal matrix

630 Index

orthonormal generalized, see generalized permutation basis, 16 matrix list, 83 maximum value of convex function, 550 set, 15 signum (sgn), 9 orthostochastic matrix permutation invariant norm, 321 and majorization, 253 permutation similar connect eigenvalues with main diagonal definition, 58 entries, 257 Weyr and Jordan forms, 206, 214 definition, 257 Perron root example of doubly stochastic, 548 definition, 525, 531 oscillatory solutions, 176 Perron vector Ostrowski set, 413 columns of the adjugate, 528 Ostrowski’s theorem, 283, 413, 416 definition, 525 Ostrowski, A., 424 irreducible nonnegative matrix, 534 Ostrowski–Brauer theorem, 424 left, 526 Ostrowski–Taussky inequality, 510 right, 525 outer product, 8 Perron’s theorem, 526 ovals of Cassini, 415 Perron, O., 524 Overton, M. L., 413 Perron–Frobenius theorem, 534 persymmetric, 36, 394 Paige, C., 162 perturbation theorem, 256 Palais matrix, 92 Piazza, G., 457 parallelogram identity Pierce, S., xii and derived norms, 318 plane rotation definition, 317 create zero entries in a vector, 94 implies triangle inequality, 318 definition, 87 partial order, 493 factor a real orthogonal matrix, 94 partial trace, 437 prove QR factorization, 94 partition PLU factorization according to columns, 17 theorem, 219 conformal, 17 Poincare separation theorem, 248 of a finite set, 17 polar coordinates, 556 of a matrix, 17 polar decomposition of a set, 16 2-by-2 matrix, 456 sequential, 16 companion matrix, 457 standard, 208 equivalent to singular value decomposition, Pearcy, C., 381 454 Pei, P., 191 explicit, 458 perfectly conditioned theorem, 449 matrix inversion, 382 polar factorization scalar multiple of an isometry, 386 normal matrix, 456 perhermitian, 36 polar form, see polar decomposition permanent, 9 polarization identity, 318 and determinant, 93 Politi, T., 458 permutation polynomial definition, 9 annihilating, 191 identity, 9 bounds on zeroes, 365 main diagonal entries, 32 degree of, 46 matrix Euclidean algorithm, 130 Birkhoff’s theorem, 548 in noncommuting variables, 119 definition, 32 monic, 46

Index 631

of a matrix, 46 positive definite on a subspace, 238 rank p(A) = 1, 199 positive matrix polynomial interpolation, 37 algebraic multiplicity of spectral radius, 526, Poore, D. E., 100 528 poorly conditioned, see ill conditioned definition, 401, 519 position operator, 130 geometric multiplicity of spectral radius, 525 positive definite function limit theorem, 526 definition, 435 Perron theorem, 526 examples, 435, 443, 444 positive eigenvector, 524 periodic, 435 positive semidefinite kernel product of two is a positive definite function, definition, 436, 478 482 Mercer’s theorem, 478 properties, 435 positive semidefinite matrix positive definite Hermitian part, 434, 436 ∗congruences, 431 positive definite kernel characteristic polynomial, 438 criterion, 482 characterizations, 438 positive definite matrix Cholesky factorization, 441 characteristic polynomial, 438 column inclusion property, 432 characterizations, 438 compression Cholesky factorization, 441 via determinant, 445 column inclusion property, 432 via principal minor sums, 445 compression via pth compound matrix, 445 via determinant, 445 via trace, 437, 445 via principal minor sums, 445 definition, 429 via pth compound matrix, 445 complex, 230 via trace, 445 real, 231 criterion on real and imaginary parts, 503 determinant, 430 definition, 429 eigenvalues, 438 complex, 230 generalized Schur complement, 437 real, 231 Gram matrix, 440, 441 determinant, 430 limit of positive definite matrices, 432 eigenvalues, 438 nonnegative eigenvalues, 230 ellipsoid, 491 nonsingular ⇔ positive definite, 431 Gram matrix, 440, 441 product of two, 486 Loewner–John matrix, 491 product with a Hermitian matrix, 486 positive eigenvalues, 230 properties, 431 positive leading principal minors, 439 real symmetric matrix, 231 product of three, 490 row inclusion property, 432 product of two, 486, 490 square root, 439 product with a Hermitian matrix, 486 square root, 2-by-2, 456 similar to a real diagonal matrix, 446 trace, 430 properties, 431 positive semidefinite on a subspace, 238 Re A−1 ! (Re A)−1, 503 positive semidefinite part of a Hermitian real symmetric matrix, 231 matrix row inclusion property, 432 best positive semidefinite approximation, 323 simultaneous diagonalization with a complex definition, 231 symmetric matrix, 487 properties, 231 square root, 439 role in Lidskii’s eigenvalue majorization square root, 2-by-2, 456 inequalities, 250 Sylvester’s criterion, 439 well-defined, 233 trace, 430 potential energy, 285

632 Index

power inequality leading numerical radius, 379 definition, 17 spectrally dominant norm, 376 eigenvalues of Jacobi matrix, 260 power method LU factorization, 216 definition, 81 moment matrix, 129 nonnegative matrix, 545 Sylvester criterion, 439 power series of matrices positive (semi)definite matrix, 430 norm convergent, 350 trailing power-bounded matrix definition, 17 characterization, 181 Sylvester criterion, 439 definition, 180 principal value of argument, 556 matrix norm and spectral radius, 368 principle of biorthogonality, see biorthogonality row- or column-stochastic matrix, 530 projection stochastic matrix, 552 definition, 38 pre-norm diagonalizable, 196 definition, 327 Drazin inverse, 189 duality theorem, 338 field of values, 380 Gelfand formula, 372 Hermitian, see Hermitian projection, 233 preorder on a set numerical radius, 380 definition, 419 orthogonal, see Hermitian projection maximal element, 419 singular values, 156, definition, 419 212, 380 primary matrix function, 350 strictly diagonally dominant, 394 prime, 176 unitary similarity canonical form, 156, 212 primitive matrix proper rotation matrix, 88, 92 alternative definition, 544 property L, 130 combinatorially symmetric, 545 property P, 130 definition, 540 property SC, see strongly connected graph index of primitivity, 542 pth root lengths of directed paths, 541 block upper triangular matrix, 130 limit theorem, 540 pure state, see density matrix location of zero entries, 545 minimal polynomial, 545 QL factorization, 91 positive main diagonal entries, 541 QR factorization positive power, 540, 542, 543 derive from Cholesky factorization, 456 powers are primitive, 541 LQ version, 90 product need not be primitive, 544 normal matrix, 155 Romanovsky’s theorem, 541 QL version, 91 principal RQ version, 91 angles, 317 theorem, 89 minor sums, 53 uniqueness, 89 definition, 52 QS factorization similarity invariant, 59 definition, 276 principal minor Kaplansky proof of rank condition, 279 definition, 17 necessary and sufficient condition for a, 276 size r,17 unitary matrix, 140 trailing quadratic form definition, 17 definition, 225 Oppenheim–Schur inequality, 509 does not determine the matrix, 232 principal submatrix simultaneous diagonalization, 490 definition, 17 quadratic inequalities determinant is a principal minor, 17 general, 498

Index 633

Hermitian–Hermitian case, 498 rank Hadamard exponential, 502 and equivalence, 13, 158, 189 Hadamard powers, 502 complex skew-symmetric matrix, 153 Hermitian–symmetric case, 502 definition, 6 Grunsky inequalities, 260, 286, 502 diagonalizable matrix, 70, 124 Hadamard exponential, 502 eigenvalues, 124, 147 Hadamard powers, 502 full-rank factorization, 13 quantum cohomology, 260 length of a longest linearly independent list of quantum mechanics, 130, 229, 305 columns, 12 quantum system length of the longest linearly independent list eigenstate, 447 of rows, 12 finite dimensional, 446 lower bound, 124, 232, 394 Heisenberg uncertainty principle, 447 nonzero eigenvalues, 124, 147 observables, 447 partitioned matrix, 20 Schrodinger¨ uncertainty principle, 447 similarity invariant, 59 Wigner–Yanase correlation, 447 singular values, 151 Wigner–Yanase skew information, 447 small perturbations increase rank, 455 quasicommute, 127, 130 triangular matrix, 124 quasidiagonal, 103 rank-nullity theorem, 6, 13, 238 definition, 32 rank-one matrix quasilinearization diagonalizable, 173, 200 representation of f (A) = (det A)1/n, 513 minimal polynomial, 200 representation of a norm as the dual of its normal, 144 dual, 339 rank-principal matrix representation of eigenvalue sums, 248 0 is a semisimple eigenvalue, 70, representation of f (A, y) = (y∗ A−1 y)−1, 475 174 representation of Schur complement, 504 definition, 20 quasitriangular, 103 EP matrix, 158, 296 definition, 32 Hermitian, 234 quasitriangularization Hermitian matrix, 20 real matrix via similarity, 103 normal matrix, 145 simultaneous, 149 nullspace A = nullspace A∗, 296 via unitary congruence, 266 skew-Hermitian matrix, 20 quaternion matrix skew-symmetric matrix, 20, 80 complex representation, 275 symmetric matrix, 20 quaternion type rank-sum inequality, 13 complex representation, 275 Rao, C. R., 476 congruent to a real matrix, 299 rational canonical form, 200 matrix, 275 Rayleigh quotient quotient property of Schur complements, 27 definition, 234 eigenvalue inclusion region, 412 radius of convergence, 350 optimal approximate eigenvalue, 412 Radjavi, H., 101, 130 theorem, 234 range Rayleigh, Lord, 238 and invariant subspace, 190 RE factorization, 308 definition, 6 real axis, 556 normal matrix, 146 real Jordan canonical form, 202 orthonormal basis for the, 453 real matrix sums, 8 ∗congruent to a, 299 ranges commuting family, 105 intersection of, 16 complex versus real orthogonal similarity, sum of, 8 275

634 Index

real matrix (cont.) regular part complex vs. real similarity, 67, 141, 275 congruence, 295 complex vs. real unitary similarity, 141 ∗congruence, see ∗congruence canonical form congruent to a, 299 regularization diagonalizable, 203 congruence, 295 negative entries of A2, 539 ∗congruence, 292 quasitriangularization, 103 regularization algorithm real eigenvalue if size is odd, 55 congruence, 295 real symetrically similar to its transpose, ∗congruence, 292 178 Reinboldt–Greub inequality, see similar to a, 202, 443 Greub–Rheinboldt inequality real normal matrix, see normal matrix, real relative gain array real orthogonal group, 85 definition, 483 real orthogonal matrix smallest eigenvalue ≥ 1, 483 canonical form, 137 relative maximum, 426 definition, 84 relative minimum, 426 Jordan canonical form, 276 residual vector, 384, 411 product of plane rotations and a reflection, 94 reversal matrix product of real Householder matrices, 93 connects forward and backward shifts, 34 real orthogonal similarity convert Hankel to Toeplitz and vice versa, of real matrices, 98 35 of real matrices that are complex definition, 33 orthogonally similar, 275 eigenvalues, 145 simultaneous, 141 reverse Fischer inequality, see Fischer real orthogonally diagonalizable, 94 inequality, reverse real part, 7, 48, 71, 476 reverse-order law, 7 real part of a complex number, 555 right eigenvector real representation definition, 77 algorithm to compute coneigenvalue, 309 if it is also a left eigenvector, 77 criterion for consimilarity, 72, 310 need not be a left eigenvector, 77 of a normal matrix, 71 right half-plane, 556 Type 1, 71 right Perron vector, see Perron vector, right 2-by-2, 148 right shift operator, 49 block 2-by-2, 187, 273, 275 ring norm, 341 Type 2, 71 Robertson’s inequality, 490 block 2-by-2, 272 Robertson, H. P., 490 real Schur form, see Schur, triangularization, Rodman, L., xii, 279, 569 real, 149 Romanovsky’s theorem, 541 real skew-symmetric matrix Romanovsky, V., 545 canonical form, 136 root real symmetric matrix conjugate pairs, 561 canonical form, 136 of an equation, 46, 561 commuting family, 145 Rosenthal, P., xiv, 75 reciprocal basis, 93 rotation matrix reduced row echelon form proper, see proper rotation matrix definition, 11 Rothblum, U., 529 leading entry, 11 row reducible operations, 9 definition, 402 type 1, 10 doubly stochastic matrix, 552 type 2, 10 family, 62 type 3, 10, 31, 219 system of linear equations, 402 rank, 12

Index 635

row inclusion property Schur form, see Schur, triangularization, definition, 216, 432 complex LU factorization, 216 Schur product, see Hadamard product positive (semi)definite matrix, 432 Schur, I., 278, 485 positive definite Hermitian part, 434 Scott, N. H., 278 positive semidefinite Hermitian part, second differences, 172 433 secondary diagonal, see counterdiagonal sufficient condition, 436 secondary eigenvalue row space upper bound, 527 definition, 8 Segre characteristic row stochastic matrix, see stochastic matrix and Jordan canonical form, 206 RQ factorization, 91 and Weyr characteristic, 173, 276 RREF, see reduced row echelon form definition, 170 selection principle, 86 Sa, E., 148 self-adjoint matrix norm Samuelson, P. N., 319 definition, 357 scalar dominates the spectral norm, 370 definition, 1 spectral norm is the only induced one, 358 matrix, 6, 30 self-adjoint norm on matrices product, see inner product definition, 357 Schatten p-norm, 465 unitarily invariant norms, 357 schlicht, 260 self-annihilating matrix Schneider, H., 370, 492 conjugate, see conjugate self-annihilating Schreiner, E., 191 matrix Schrodinger¨ uncertainty principle, 447 Jordan canonical form, 172 Schur unitary similarity canonical form, 157, complement 213 and partitioned forms for the inverse, 18 semi-inner product basic ∗congruence identity, 495 Cauchy–Schwarz inequality, 316 definition, 25 covariance of X and Y in the state R, 446 generalized, 437, 453 definition, 15, 316 Haynsworth’s theorem, 298 equality in the Cauchy–Schwarz inequality, of a strictly diagonally dominant matrix, 317 395 induces a seminorm, 316 positive semidefinite matrix, 437 semiconvergent matrix quotient property, 27 definition, 191 variational characterization, 504 representation via the Drazin inverse, 191 inequality, 102, 144, 156, 323 semilinear transformation, 305 for A2, 158 seminorm

for C2(A), 158 deﬁnition, 314 strengthened, 158, 323 induced by a semi-inner product, 316 strengthened further, 158, 323 null space of a, 317 majorization theorem, 249, 259 on matrices, 341 norm, 341 seminorm on matrices Oppenheim inequality, see Oppenheim–Schur similarity invariant, 378 inequality semisimple eigenvalues, see eigenvalues, product theorem, 479 semisimple real representation, 272 nonnegative matrix, 530 triangularization separating hyperplane theorem, 558 complex, 101 separation theorem real, 103 bordered Hermitian matrices, 242 uniqueness, 113 Poincare, 248

636 Index

Sergeichuk, V. V., xiv, 148, 215, 278, 279, 300, by congruence 311, 569 via nonsingular, 287 sesquilinear via unitary, 286 form by ∗congruence, 296 absolute value almost determines the by congruences, 485, 487 matrix, 232 via nonsingular, 287 determines the matrix, 232 commuting family form generated by a Hermitian matrix, 227 complex, 64 function, 15 real, 68 inner product, 315 commuting Hermitian matrices, 229 semi-inner product, 316 definition, 61, 64 Shapiro, H., 215 unitary, 135 Shen, J., 545 via unitary equivalence, 154, 155 Sherman–Morrison–Woodbury formula, 19 simultaneous similarity shift A and B; A∗ and B∗, 457 backward and forward, 34 Jordan form, 214 Shisha, O., 429 Weyr form, 210 Siegel, C. L., 278 simultaneous singular value decomposition, see sign pattern matrix singular value decomposition, requires distinct eigenvalues, 129, 130 simultaneous signature, 282 symmetric matrices, 157 signum (sgn) of a permutation, 9 simultaneous triangularization similarity 2-by-2 block upper triangular matrices, 128 canonical blocks, 167 commuting family, 103, 115, 116 complex orthogonal, 272, 275 little McCoy theorem, 127 definition, 41, 58 McCoy’s theorem, 119 different basis representations of a single necessary and sufficient conditions linear transformation, 43 n = 2, 127 equivalence relation, 58 n = 3, 127 invariant, 167, 275, 378 simultaneous unitary congruence matrix, 58 and unitary similarity, 278 matrix and its transpose, 177, 180, 271 block matrix criterion, 278, 279, 455 permutation, 58 simultaneous unitary similarity real, 67, 141, 275, 300 and unitary congruence, 277, 278 real orthogonal, 141, 275 block matrix criterion, 279 to a real matrix, 202, 229 criterion for, 147 to a symmetric matrix, 271 definition, 147 to conjugate transpose, 229 verify with finitely many computations, 147 uniqueness to diagonalize, 67 simultaneous unitary triangularization, 118 unitary singular matrix definition, 83, 94 best approximation by a, 461 simple directed cycle, see cycle, simple directed unique, 461, 476 simple eigenvalue, see eigenvalues, simple definition, 14 simultaneous condiagonalization, 307 distance to a nearest, 369, 461 simultaneous congruence, 286, 568 nearest, 369 simultaneous ∗congruence, 286, 568 singular part simultaneous diagonalization congruence, 295 via unitary congruence and ∗congruence , 286 ∗congruence, see ∗congruence canonical form by ∗congruence, 487 singular value decomposition via nonsingular, 287, 294 AA¯ = 0, 157 via unitary, 286 AA¯ is positive semidefinite, 277, 311

Index 637

adj(A), 155 skew information, see Wigner–Yanase skew coninvolutory matrix, 278, 311 information equivalent to polar decomposition, 454 skew orthogonal, 92 normal matrix, 156 skew perhermitian, 36 real matrix, 154 skew persymmetric, 36 self-annihilating matrix, 157 skew sum, 568 simultaneous, 155, 157 skew symmetric skew-symmetric matrix, 153, 270 definition, 7 symmetric matrix, see symmetric matrix, skew-Hermitian matrix singular value decomposition definition, 7 theorem, 150 rank-principal, 20 thin, 443, 450, 456 skew-Hermitian part Toeplitz matrix, 274 definition, 7, 227 uniqueness of factors, 152, 156 Toeplitz decomposition, 227, 289, 567 distinct singular values, 159 skew-symmetric matrix symmetric matrices, 159, 263 canonical pairs, 567 singular values even rank, 153 2-by-2 block unitary matrix, 156 Jordan canonical form, 276 2-by-2 matrix, 151 normal, 146 adj(A), 155 product of two, 276 and eigenvalues, 347 product with symmetric, 276 bidiagonal matrix, 259 rank 1, 157 companion matrix, 197 rank-principal, 20 continuity, 152 real definition, 151 canonical form, 136 derivative, 453 singular value decomposition, 153, 270 inclusion set, 424, 455 unitary, 139 interlacing, 451, 457 unitary congruence canonical form, 270 multiplicity, 151 skew-symmetric part nilpotent bidiagonal block, 151 definition, 7, 567 normal eigenvector, 158 Smiley, M. F., 324 normal matrix, 156 So, W., xiv of A, A¯, AT ,andA∗, 151 Sorensen, D., 175 perturbation bounds, 451 span, 2 product and determinant, 151 Specht’s theorem rank, 151 approximate version, 101 simple, 151 statement, 97 skew-symmetric matrix, 153 Specht, W., 100 sum of squares, 151 special relativity, 300 trace norm, see trace norm spectral characteristic tridiagonal matrix, 259 convex function, 378 unitary matrix, 155 definition, 378 variational characterization, 451 minimally spectrally dominant, 378 Weyl product inequalities, 370, spectral decomposition, 134 454 spectral matrix, 368, 552 singular vectors spectral norm definition, 453 definition, 346 left and right, 453 dominated by any self-adjoint matrix norm, sip matrix, see reversal matrix 370 skew centrohermitian, 36 dual of trace norm, 362 skew centrosymmetric, 36, 200 induced and unitarily invariant, 357

638 Index

spectral norm (cont.) of a diagonalizable matrix, 69 induced by Euclidean norm, 346 of a positive (semi)definite matrix, 439, 443 monotone on nonnegative matrices, 368 of a symmetric unitary matrix, 140 not absolute, 368 squared-normal matrix only self-adjoint induced norm, 358 canonical form, 147, 148 unitarily invariant, 346 definition, 147 spectral radius standard basis, see basis, standard 2-by-2 Hermitian, 233 standard inner product, see inner product, compound matrix, 108 standard definition, 52 standard partition, 208 equal to a matrix norm, 368 standardized norm Gelfand formula, 349 absolute, 339 greatest lower bound for all matrix norms, definition, 339 347 dual, 339 T Ln − Ln , 198 Steinitz theorem, see Caratheodory–Steinitz´ not subadditive, 118 theorem subadditive on commuting matrices, 117 Stenzel, H., 279 submultiplicative on commuting matrices, Stephens, A. B., 533 117 stochastic matrix upper bound, 390, 391, 392 column, 547 upper bound by any matrix norm, 347 compact convex set, 548 spectral theorem definition, 547 Hermitian matrix, 135, 229 power bounded, 530 normal matrix, 133 power-bounded, 552 real symmetric matrix, 234 semigroup, 551 uniqueness, 134 semisimple eigenvalues, 552 spectrally dominant norm Stone, B. J., 370 and compatible vector norm, 347 Strang, G., 191, 370 characterization, 377 strict contraction, see contraction, strict compatible with a norm on Cn, 374 strict diagonal dominance, 352 definition, 373 strictly diagonally dominant maximum-modulus eigenvalues are block, 395 semisimple, 376 definition, 392 noncompatible example, 375 idempotent, 394 spectrum positive (semi)definite matrix, 438 definition, 45 preserved under Gaussian elimination, 394 nonempty, 47 Schur complement, 395 speed of light, 300 sign of determinant, 394 spread strong majorization, 248 2-by-2 Hermitian, 233 strongly connected graph, 398, 400, 418 definition, 147 Strutt, John William, see Rayleigh, Lord lower bound, 258 subadditivity, 314 upper bound, 147 subdiagonal, 5 square root sublist, 3 2-by-2 positive semidefinite matrix, 456 submatrix block upper triangular matrix, 130 definition, 5 concave on positive definite matrices, 505 notation, 17 definition, 69 principal, 17 inverse square root is convex on positive submultiplicativity, 341 definite matrices, 505 subordinate norm, see compatible norms monotone on positive definite matrices, 495, subspace 501, 505 coninvariant, 262

Index 639

definition, 2 2-by-2 matrix, 264 intersection lemma, 4, 235, 238 definition, 7, 567 invariant, 62, 262 unitarily invariant norm bound, 476 nontrivial, 2, 62 Symons, J. S. V., 320 proper, 2 Szasz´ inequality sum, 2 refines Hadamard’s inequality, 508 trivial, 2, 62 theorem, 508 sum norm, see or Sze, N. S., 75 sum of subspaces, 2 superdiagonal, 5 Tadmor, E., 381 SVD, see singular value decomposition Takagi Sylvester factorization, see Autonne–Takagi criterion for positive definite matrix, 439 factorization equation, 111 theorem, see Autonne–Takagi factorization identity for bordered determinants, 27 Takagi, T., 278 rank inequality, 13 Tan, C., 529 Sylvester’s law of inertia Taussky’s inequality, 490 analog for congruence, 284 Taussky’s theorem, 404

generalization to Mn, 289 Taussky, O., 130, 396, 429, 490 quantitative form, 283 Taussky–Ostrowski inequality, see singular congruences, 284 Ostrowski–Taussky inequality theorem, 282 Tener, J. E., 100, 273 Sylvester’s theorem theorem of the alternative, see Fredholm, law of inertia, see Sylvester’s law of inertia alternative linear matrix equations, 111, 127 thin singular value decomposition, see singular symmetric gauge function value decomposition, thin definition, 464 thin SVD, see singular value decomposition, Loewner–John matrix of a, 492 thin unitarily invariant norms on matrices, 465 Thompson, R. C., 569 symmetric matrix time reversal, 305 2-by-2 unitary similar to a, 162 Toeplitz decomposition, 7, 227, 289, 567 A = SST , 272 Toeplitz matrix canonical pairs, 567 definition, 34, 427 definition, 7 eigenvalues, 394 diagonalizable, 272, 277 persymmetric, 36 every matrix similar to a, 271, 272 reversal similar to transpose, 34, 177, 270 involution, 274 singular value decomposition, 274 Jordan block unitarily similar to a, 271 tridiagonal normal, 146 nearly normal, 144 orthogonally diagonalizable, 272, 277 normal, 144 product of two, 178, 271, 274 Toeplitz moment sequence, 427 product of two real, 274 trace product with Hermitian, 306 definition, 7 product with skew symmetric, 276 moments and eigenvalues, 126 rank-principal, 20 negative definite matrix, 431 simultaneous SVD, 157 norm, see trace norm singular value decomposition, 153, 156, 159, of AA∗,7 263, 272, 275, 308 partial, 437 unitary, 139 positive (semi)definite matrix, 430 square root, 140 similarity invariant, 59 unitary congruence, 153 Specht identities, 97 symmetric part sum of eigenvalues, 50

640 Index

trace (cont.) eigenvalues, 82, 222 theorem, see von Neumann, trace theorem distinct real, 174 tr A−1 is strictly convex on positive definites, real, 174 489 zero main diagonal, 81, 174 trace norm Hermitian and irreducible/unreduced definition, 362 distinct eigenvalues, 258 dual of spectral norm, 362 eigenvalue interlacing, 258 upper bound on tr Am , 539 Jacobi matrix, see Jacobi matrix trailing principal minor, see principal minor, LU factorization, 222 trailing nearly normal, 144 trailing principal submatrix, see principal normal, 144, 147 submatrix, trailing real eigenvalues, 232 transpose singular values, 259 2-by-2 matrix is unitarily similar to its, 99 Toeplitz definition, 6 eigenvalues, 82 matrix ∗congruent to its, 299 tridiagonalization matrix congruent to its, 299 Lanczos, see Lanczos tridiagonalization matrix not unitarily similar to its, 99 tripotent, 196 matrix similar to its, 177, 180, 190, 191 trivial cycle, see cycle, trivial real matrix is real symmetric similar to its, trivial linear combination 178 definition, 2 transposition, 32 Tsai, Y. L., xvi trial and error, 309 type 1 row operations, see row, operations, triangle inequality type 1 complex numbers type 2 row operations, see row, operations, case of equality, 96, 520 type 2 proof and case of equality, 556 type 3 row operations, see row, operations, equality in the, 520 type 3 norms, 314 triangular equivalence uncoupled equations, 176, 285 canonical form, 220 uniformly continuous function, see continuous complete set of invariants, 220 function, uniformly definition, 41, 220 unistochastic matrix triangular matrix and Hoffman–Wielandt theorem, 407 block, 32 connect eigenvalues with diagonal entries, commutes with, 31 257 definition, 31 definition, 257 essentially, 32 doubly stochastic but not unistochastic, strictly, 31 413 unit, 31 example of doubly stochastic, 548 triangularization unit real matrix, see Schur, triangularization, real lower triangular matrix, see lower triangular sequential deflation, 101 matrix, unit simultaneous, see simultaneous triangular matrix, see triangular matrix, unit triangularization, 149 upper triangular matrix, see upper triangular tridiagonal matrix matrix, unit block tridiagonal, 36 unit ball correlation matrix compact, 336 eigenvalues, 445 convex, 337 definition, 35 definition, 335 determinant, 35, 222 equilibrated, 336

Index 641

geometric characterization, 337 skew-symmetric matrix, 153 not compact, 337 symmetric matrix, 159, 263 unit disc in complex plane, 556 to a real matrix, 273 unit triangular equivalence verify with finitely many calculations, 278 canonical form, 221 Youla’s theorem, 266 definition, 221 unitary congruence canonical form unit vector, see vector, unit 2-by-2 matrix, 264 unital matrix norm AA¯ is Hermitian, 277 definition, 344 AA¯ is positive semidefinite, 311 not induced, 357 congruence-normal matrix, 277 unitarily diagonalizable coninvolutory matrix, 311 definition, 94 conjugate-normal matrix, 268, 274 unitarily invariant matrix norm conjugate-self-annihilating matrix, 157 characterization, 469 skew-symmetric matrix, 153, 270 convex set, 363, 469 symmetric matrix, 153 definition, 357 unitary matrix, 270 dominates the spectral norm, 370 unitary consimilarity, see unitary congruence spectral norm is the only induced, 357 unitary dilation, 162 unitarily invariant norm on matrices unitary equivalence definition, 357 definition, 41, 83 scalar multiples of Frobenius norm are only equality of Frobenius norms is a necessary absolute, 470 condition, 94 symmetric gauge functions, 464 invariants for, 155 unitarily invariant vector norm involves two independent unitary matrices, Euclidean norm, 84 149 only scalar multiples of Euclidean norm, 320, simultaneous 322 to diagonal matrices, 154 permutation invariant norm, see permutation to real or nonnegative diagonal matrices, invariant norm 155 unitarily irreducible, 233 to real or nonnegative diagonal matrices unitarily reducible, 233 via real orthogonal equivalence, 155 unitary unitary matrix definition, 84 ∗congruence, 457 diagonalization 2-by-2 block matrix, 87, 156 normal, 133 2-by-2 unitarily similar to complex uniqueness, 134 symmetric, 162 dilation, 162 commutes with every, 130 equivalence, 156 compact set, 86 group, 85 cramped, 146 orthogonal, see orthogonal, unitary definition, 7 Weyr form, 211 diagonal, 91 commutes with, 207 product of Householder matrices, 93 unitary congruence QS factorization, 140 block matrix criterion, 278, 279, 455 singular values of blocks, 156 characterization, 277 skew-symmetric, 139 criterion, 278, 455 SQ factorization, 140 definition, 41 symmetric, 139, 140 is unitary consimilarity, 300 unitary congruence canonical form, 270 partial triangularization by a, 261 unitary similarity simultaneous, see simultaneous unitary criterion, 97 congruence definition, 41, 83, 94

642 Index

unitary similarity (cont.) vec mapping of real matrices, 98 definition, 21 simultaneous, see simultaneous unitary vector similarity all ones, 8, 57 simultaneous similarity: A and B; A∗ and B∗, column, 2 457 coordinate representation of, 39 simultaneous, of real matrices, 141 isotropic, see isotropic vector Specht’s theorem, 97 normalized, 15, 45, 83, 314 to a complex symmetric matrix, 271, 273 unit, 15, 314 to a matrix with equal diagonal entries, 95 zero, 1 to a real matrix, 273 vector norm, 314 to an upper Hessenberg matrix, 96 vector norm on matrices to transpose, 99, 273 definition, 341 verify with finitely many computations, Gelfand formula, 372 98 scale to a matrix norm, 373 unitary similarity canonical form vector space self-annihilating matrix, 213 complete, 329 unitary similarity invariant complex, 2 definition, 379 definition, 1 univalent, 260 real, 2 unreduced subspace, 2 Hessenberg matrix, see Hessenberg matrix, zero, 2 unreduced vectors upper half-plane, 556 angle between two real, 15, 473 upper triangular matrix Vinsonhaler, C., xiv, 211, 215 block, 32 volume commutes with, 31, 113 and determinant, 491 definition, 31 von Neumann Jordan block, 164 trace theorem, 458, 550 Jordan canonical form, 167 case of equality, 460 LPU factorization, 219 von Neumann, J., 320, 340 LU factorization, 216 not square, 31 weak minimum principle, 480 PLU factorization, 219 weakly connected graph, 418 Schur triangularization, 101 weakly irreducible matrix unit, 31, 218 characterization, 418 unitary Weyr form, 211 definition, 418 usual inner product, see inner product, usual weakly monotone norm, 340 Wedderburn rank-one reduction, 14 van den Driessche, P., xii, 457 Wei, M., 162 Vandermonde matrix Wei, Y., 279 and the moment matrix, 128 Weierstrass theorem, 565 definition, 37 well conditioned inverse of, 37 matrix inversion, 382 Varga, R., 396 Weyl product inequalities, 370, 454 variance Weyl’s theorem definition, 319 eigenvalues of A + B, 239, 260 variance of X in the state R interlacing, 241 definition, 446 monotonicity, 241 not smaller than Wigner–Yanase skew singular values of A + B, 454 information, 448 Weyl, H., 239

Index 643

Weyr block, 204, 205 real part is a semi-inner product on Weyr canonical form Hermitians, 448 commuting family, 211 skew information definition, 206 definition, 447 Jordan block, 206 nonnegative on normal matrices, 448 nonderogatory matrix, 206 not greater than variance, 448 theorem, 205 Wimmer, H., 457 unitary, 211 Woerdeman, H., xiv unitary, commutes with, 207 Wolkowicz, H., 148 Weyr characteristic Woodbury, see Sherman–Morrison–Woodbury AB versus BA, 191 formula and concanonical form, 305 word and Segre characteristic, 173, 276 definition, 97 and Weyr canonical form, 206 degree of a, 97 definition, 170 in A and A∗,97 square of a nilpotent matrix, 310 Wronskian, 56 Weyr matrix commutes with, 208, 211 Youla’s factorization, see Youla’s theorem definition, 204 Youla’s theorem, 266 Weyr, E., 215 Youla, D. C., 278 Wiegmann, N. A., 130 Young diagram, see dot diagram Wielandt Young tableau, 214 inequality, 471 angle between Ax and Ay with x ⊥ y, 473 zero equivalent to Kantorovich inequality, 472 matrix, see matrix, zero matrix, 158, 368, 455, 544 of a function, 46 singular values and eigenvalues, 450, vector, see vector, zero 457 zero-one matrix theorem on index of primitivity, 543 normal, 148 Wielandt, H., 260, 413, 543 not permutation similar, 174 Wigner–Yanase Zhan, X., 191 correlation Zhang, F., xiv, 279 definition, 447 Zhang, X., 424 not a semi-inner product, 448 Zhang, X. D., 515