ASummary of Notation

Re(z), 7im(z), 7z, 8Q(

√p), 8

Z/(p), 8GF(p), 8∏

i∈Ω Vi , 23Mk×n(V ), 24V Ω , 23χB , 24∐

i∈Ω Vi , 26V (Ω), 26Fv, 26C(a, b), 26FD, 27affh(D), 29∑

i∈Ω Wi , 30K−, 42v × w, 42K+, 43deg(f ), 44F [X], 44F [g(X)], 45Φq(X), 48μ(d), 48F [X1, . . . ,Xn], 50dim(V ), 71U ⊕ W , 74⊕

i∈Ω Wi , 74gr(f ), 91Hom(V ,W), 92ker(α), 94im(α), 94

AT , 95Aff(V ,W), 96∼=, 97rk(α), 98null(α), 98V/W , 110End(V ), 113σc, 113Aut(V ), 115εhk , 116εh;c , 116εhk;c , 116ΦBD , 133v � w, 136v ∧ w, 136Ehk , 153Eh;c , 153Ehk;c , 153Sn, 224sgn(π), 224|A|, 225adj(A), 235spec(α), 255ρ(A), 258comp(p), 265A ∼ B , 266Ann(v), 272mA(X), 274F [α]v0, 297LR(V ), 298D(V ), 317tr(A), 318〈v,w〉, 333

J.S. Golan, The Linear Algebra a Beginning Graduate Student Ought to Know,DOI 10.1007/978-94-007-2636-9, © Springer Science+Business Media B.V. 2012

479

480 A Summary of Notation

v · w, 334DH , 334‖v‖, 338eA, 353cos(A), 357sin(A), 357v ⊥ w, 369W⊥, 374α∗, 383Rα , 400SO(n), 424√

α, 430α+, 442Bil(V × W,Y), 453

f op , 453Bil(V × W), 455TBD(f ), 456v ⊥f w, 458A⊥f , 458q(v), 461V ⊗ W , 465v ⊗ w, 465A ⊗ B , 467V ∧ V , 470V ⊗k , 470∧k

V , 470∧

(V ), 470

BIndex to Thumbnail Photos

ANiels Henrik Abel, 15, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachRobert Adrain, 448, Edgar Fahs Smith Collection; reproduced with the kind permis-

sion of the University of Pennsylvania LibrariesA. Adrian Albert, 44, courtesy of the American Mathematical SocietyShimshon Amitsur, 238, source unknownGeorge Andrews, 230, with kind permission of George E. AndrewsApollonius of Perga, 363, source unknownJean-Robert Argand, 7, Bibliothèque nationale de FranceEmil Artin, 334, Archives of the Mathematisches Forschungsinstitut Oberwolfach

BStefan Banach, 342, Stefan Banach homepage http://banach.univ.gda.pl/Erwin Bareiss, 236, with kind permission of The Daily NorthwesternGiusto Bellavitis, 7, source unknownEugenio Beltrami, 432, source unknownAdi Ben-Israel, 445, with kind permission of Adi Ben-IsraelSergei Natanovich Bernstein, 105, Archives of the Mathematisches Forschungsin-

stitut OberwolfachWilhelm Bessel, 381, with kind permission of the Leibniz-Institut für Astrophysik

PotsdamÉtienne Bézout, 471, source unknownJacques Philippe Binet, 139, source unknownGeorge D. Birkhoff, 148, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachAndreas Blass, 69, courtesy of A. BlassMaxime Bôcher, 151, Harvard University Archives, HUPBernard Bolzano, 23, with kind permission of Frommann–Holzboog PublishersEmile Borel, 239, source unknownRaoul Bott, 72, Archives of the Mathematisches Forschungsinstitut OberwolfachAlbert H. Bowker, 345, The City University of New YorkAlfred Brauer, 352, Archives of the Mathematisches Forschungsinstitut Oberwol-

fach

J.S. Golan, The Linear Algebra a Beginning Graduate Student Ought to Know,DOI 10.1007/978-94-007-2636-9, © Springer Science+Business Media B.V. 2012

481

482 B Index to Thumbnail Photos

Viktor Yakovlevich Bunyakovsky, 337, source unknownWilliam Burnside, 280, source unknown

CGiovanni Domenico Cassini, 352, source unknownEugène Catalan, 173, © Collections artistiques de l’Université de Liège; reprinted

with kind permissionAugustin-Louis Cauchy, 227, source unknownArthur Cayley, 133, courtesy of Special collections, Fine Arts Library, Harvard Uni-

versityPafnuty Lvovich Chebyshev, 371, source unknownClaude Chevalley, 465, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachAndré-Louis Cholesky, 407, Collections École polytechnique (SABIX)William Kingdon Clifford, 66, courtesy of the London Mathematical SocietyRichard Courant, 401, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachGabriel Cramer, 237, source unknown

DJean d’Alembert, 256, source unknownRichard Dedekind, 6, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachAbraham de Moivre, 298, source unknownAugustus DeMorgan, 6, source unknownRené Descartes, 7, Harvard Arts MuseumLeonard Dickson, 40, courtesy of the American Mathematical SocietyJean Dieudonné, 269, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachPaul Dirac, 322, source unknownCharles Lutwidge Dodgson, 230, source unknown

EArthur Eddington, 139, source unknownFerdinand Eiseinstein, 48, source unknownLeonhard Euler, 7, ETH-Bibliothek Zurich, Image Archive

FDimitri Konstantinovich Fadeev, 203, with kind permission of Sergei VostokVera Nikolaevna Fadeeva, 203, with kind permission of Dr. Vera SimonovaLeonardo Fibonacci, 299, source unknownErnst Fischer, 401, Archives of the Mathematisches Forschungsinstitut OberwolfachJoseph Fourier, 153, Smithsonian InstitutionJohn G.F. Francis, 379, with kind permission of Frank UhligMaurice Fréchet, 317, Archives of the Mathematisches Forschungsinstitut Oberwol-

fach

B Index to Thumbnail Photos 483

Ivar Fredholm, 326, source unknownGeorg Frobenius, 271, Archives of the Mathematisches Forschungsinstitut Ober-

wolfach

GGalileo Galilei, 23, © National Maritime Museum, Greenwich, LondonEvariste Galois, 9, Bibliothèque nationale de FranceCarl Friedrich Gauss, 196, Archives of the Mathematisches Forschungsinstitut

Oberwolfach © Universität Göttingen, Sammlung SternwarteIsrail Moisseevich Gelfand, 450, Archives of the Mathematisches Forschungsinsti-

tut OberwolfachSemyon Aranovich Gershgorin, 349, source unknownJosiah Willard Gibbs, 22, Office of Public Affairs, Yale University, 1879–1989 (in-

clusive). Manuscripts & Archives, Yale UniversityJames Wallace Givens, 420, with kind permission of Walter GanderGene H. Golub, 432, Wikimedia commonsJorgen Gram, 336, source unknownHermann Grassman, 22, source unknownThomas Greville, 445, with kind permission of Mrs GrevilleAlexandre Grothendieck, 345, Archives of the Mathematisches Forschungsinstitut

OberwolfachLouis Guttman, 155, with kind permission of Nurit Guttman

HAlfréd Haar, 376, source unknownJacques Hadamard, 317, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachHans Hahn, 342, source unknownGeorg Hamel, 69, courtesy of Professor Richard von MisesRichard Hamming, 352, Special Collections & Archives, Dudley Knox Library,

Naval Postgraduate SchoolHermann Hankel, 241, source unknownFelix Hausdorff, 68, Hausdorff Research Institute for MathematicsOliver Heaviside, 22, © the estate of Oliver Heaviside, reproduced with kind per-

mission of Alan HeatherKurt Hensel, 274, Archives of the Mathematisches Forschungsinstitut OberwolfachCharles Hermite, 73, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachKarl Hessenberg, 473, with kind permission of Karl Hessenberg’s daughter, Brigitte

BossertMagnus Hestenes, 301, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachDavid Hilbert, 160, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachGeorge William Hill, 189, courtesy of the American Mathematical Society

484 B Index to Thumbnail Photos

Otto Hölder, 345, UAL, FS N 191Heinz Hopf, 72, Archives of the Mathematisches Forschungsinstitut OberwolfachAlston Householder, 423, source unknownLoo-Keng Hua, 12, source unknownAdolph Hurwitz, 454, ETH-Bibliothek Zurich, Image Archive

JCarl Gustav Jacob Jacobi, 205, Archives of the Mathematisches Forschungsinstitut

OberwolfachNathan Jacobson, 44, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachCamille Jordan, 432, source unknownPascual Jordan, 44, the estate of Friedrich Hund, with kind permission of Gerhard

HundWilhelm Jordan, 196, source unknown

KWilliam Kahan, 432, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachAnatoli Alexeevich Karatsuba, 46, with kind permission of Ekatherina KaratsubaSamuel Karlin, 271, source unknownWilhelm Karl Joseph Killing, 42, source unknownHelmut Kneser, 49, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachLeopold Kronecker, 6, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachAlexei Nikolaevich Krylov, 297, source unknownVera Kublanovskaya, 379, with kind permission of Vera Kublanovskaya

LJoseph-Louis Lagrange, 163, ETH-Bibliothek Zurich, Image ArchiveEdmund Laguerre, 191, source unknownCornelius Lanczos, 301, source unknownAdrien-Marie Legendre, 371, Bibliothèque de l’Institut de FranceTullio Levi-Civita, 465, source unknownYaakov Levitzki, 238, with kind permission of Alexander LevitzkiJoseph Liouville, 256, source unknownDudley Littlewood, 42, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachLiu Hui, 191, source unknownKarl Loewner, 404, source unknownAlfred Loewy, 266, source unknownHendrick Antoon Lorentz, 463, Museum Boerhaave Leiden

B Index to Thumbnail Photos 485

MCyrus C. Macduffee, 266, © Mathematical Association of America 2011; all rights

reservedColin Maclaurin, 237, Maclaurin, C., A Treatise of Fluxions, T.W. and T. Ruddi-

mans, 1742Andrei Andreyevich Markov, 151, source unknownJames Clerk Maxwell, 22, courtesy of Special collections, fine Arts Library, Harvard

UniversityWilliam Henry Metzler, 354, source unknownJohn Milnor, 72, Archives of the Mathematisches Forschungsinstitut OberwolfachHermann Minkowski, 340, ETH-Bibliothek Zurich, Image ArchiveCleve Moler, 380, courtesy of The MathWorks, Inc.Eliakim Hastings Moore, 9, courtesy of the American Mathematical Society

NMark Aronovich Naimark, 450, courtesy of the American Mathematical SocietyAlexander Ivanovich Nekrasov, 206, source unknownRolf Nevanlinna, 403, source unknownEmmy Noether, 40, Bryn Mawr College Library, Special Collections

OAndrew Odlyzko, 301, with kind permission of Andrew OdlyzkoAlexander Markovich Ostrowski, 47, Archives of the Mathematisches Forschungsin-

stitut Oberwolfach

PLuca Pacioli, 299, source unknownHenri Padé, 239, source unknownGiuseppe Peano, 67, Department of Mathematics, University of Torino, Italy, www.

peano2008.unito.itBenjamin Peirce, 40, Harvard University Archives, HUPCharles Sanders Peirce, 40, Harvard University Archives, HUPRoger Penrose, 441, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachOskar Perron, 271, Archives of the Mathematisches Forschungsinstitut OberwolfachJohann Pfaff, 228, source unknownGeorg Pick, 403, source unknownSalvadore Pincherle, 89, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachJules Henri Poincaré, 189, The University of Chicago Yerkes Observatory, courtesy

AIP Emilio Segre Visual Archives, Physics Today Collection and Tenn Collec-tion

RGregorio Ricci-Curbastro, 465, source unknownFrigyes Riesz, 383, source unknownHeinz Rutishauser, 379, with kind permission of Walter Gander

486 B Index to Thumbnail Photos

SYousef Saad, 375, with kind permission of Y. SaadJean Claude Saint-Venant, 22, source unknownHeinrich Scherk, 227, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachErhardt Schmidt, 26, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachIsaac Jacob Schoenberg, 56, Archives of the Mathematisches Forschungsinstitut

OberwolfachOtto Schreier, 334, source unknownMartin Schultz, 375, with kind permission of Martin SchultzIssai Schur, 161, Archives of the Mathematisches Forschungsinstitut OberwolfachHermann Schwarz, 337, ETH-Bibliothek Zurich, Image ArchiveTakazaku Seki Kowa, 191, source unknownKenjiro Shoda, 320, source unknownThomas Simpson, 187, source unknownHenry Smith, 462, source unknownSergei Lvovich Sobolev, 359, Archives of the Mathematisches Forschungsinstitut

OberwolfachRichard V. Southwell, 207, Neville Miles, Imperial College LondonErnst Steinitz, 6, Archives of the Mathematisches Forschungsinstitut OberwolfachG. W. Stewart, 380, with kind permission of Eric de SturlerEduard Stiefel, 301, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachMarshall Stone, 425, courtesy of the American Mathematical SocietyVolker Strassen, 166, with kind permission of Volker StrassenJohn William Strutt (Lord Rayleigh), 401, Science Photo LibraryCharles-François Sturm, 256, source unknownPeter Swinnerton-Dyer, 169, with kind permission of Sir Peter Swinnerton-DyerJames Joseph Sylvester, 24, source unknown

THenry Taber, 318, Clarke University ArchivesWilliam Guthrie Tait, 66, courtesy of Special collections, Fine Arts Library, Harvard

UniversityJohn Tate, 280, Archives of the Mathematisches Forschungsinstitut OberwolfachOlga Taussky-Todd, 349, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachAngus Taylor, 72, with kind permission of UC BerkeleyWilliam Thurston, 271, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachOtto Toeplitz, 203, Archives of the Mathematisches Forschungsinstitut OberwolfachAlan Turing, 169, National Portrait Gallery

B Index to Thumbnail Photos 487

VAlexandre-Théophile Vandermonde, 163, source unknownFerdinand von Lindemann, 73, Archives of the Mathematisches Forschungsinstitut

OberwolfachRichard von Mises, 281, Archives of the Mathematisches Forschungsinstitut Ober-

wolfachJohn von Neumann, 210, courtesy of the American Mathematical Society

WHeinrich Weber, 6, Archives of the Mathematisches Forschungsinstitut OberwolfachJoseph Henry Wedderburn, 40, source unknownKarl Weierstrass, 455, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachHermann Weyl, 71, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachHassler Whitney, 57, The Shelby White and Leon Levy Archives Center, Institute

for Advanced Study, Princeton, NJ, USA; photographer Herman LandshoffNorbert Wiener, 342, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachJames Wilkinson, 281, courtesy of NPLShmuel Winograd, 166, with kind permission of the Department of Computer Sci-

ence, City University of Hong KongErnst Witt, 462, Archives of the Mathematisches Forschungsinstitut OberwolfachJosef Wronski, 229, source unknown

YDavid M. Young, Jr., 207, Archives of the Mathematisches Forschungsinstitut Ober-

wolfach

ZHans Zassenhaus, 49, Archives of the Mathematisches Forschungsinstitut Oberwol-

fachMax Zorn, 68, with kind permission of Jens Zorn

Index

AAddition

in a field, 5vector, 21

Adjacency matrix, 399Adjoint matrix, 235Adjoint transformation, 383Affine combination, 29Affine hull, 29Affine subset, 96Affine transformation, 96Algebra, 39

anticommutative, 39associative, 39baric, 318Baxter, 90Cayley, 336commutative, 39convolution, 41division, 52entire, 44exterior, 470Jordan, 43Lie, 41optimization, 9quaternion, 66tensor, 470unital, 39

Algebraic, 72Algebraic degree, 110Algebraic multiplicity, 270Algebraically closed, 49Almost positive semidefinite, 415Alphabet, 23Amitsur–Levitzki Theorem, 238Angle, 369Annihilate, 272Anti-Hermitian matrix, 413Anticommutative algebra, 39Antitrace, 329

Apollonius’ identity, 363Arnoldi process, 375Associative algebra, 39Automorphism, 115

elementary, 115of algebras, 115unitary, 419

Axiom of Choice, 2

BBand matrix, 149Baric algebra, 318Basis, 63

canonical, 64, 300dual, 323Hamel, 69

Baxter algebra, 90Bernstein function, 105Bessel’s identity, 380Best approximation, 447Bijective, 2Bilinear form, 455

nondegenerate, 455Bilinear transformation, 453

skew-symmetric, 453symmetric, 453

Binet–Cauchy identity, 361Binomial formula, 17Block form, 138Bounded, 67, 340Bounded support, 393Bra-ket product, 136Brauer’s Theorem, 351Bunch–Kaufman algorithm, 408Burnside’s Theorem, 279

CCanonical basis, 64, 300Cantor set, 69Cartesian product, 2Cassini oval, 351

J.S. Golan, The Linear Algebra a Beginning Graduate Student Ought to Know,DOI 10.1007/978-94-007-2636-9, © Springer Science+Business Media B.V. 2012

489

490 Index

Cauchy–Schwarz–Bunyakovsky Theorem, 337Cayley algebra, 336Cayley representation, 440Cayley–Hamilton Theorem, 276Chain, 61Chain subset, 61Change-of-basis matrix, 162Characteristic, 9Characteristic function, 24Characteristic polynomial, 264, 298

generalized, 282Characteristic value, 255Characteristic vector, 255Chebyshev polynomial, 370Cholesky decomposition, 406Circuit, 71Circulant matrix, 172Co-independent hyperplanes, 332Coefficient, 44

Fourier, 378leading, 44Taylor, 117

Coefficient matrix, 191Column equivalent, 161Column space, 199Combination

affine, 29linear, 27uniform, 38zero-sum, 38

Commutative algebra, 39Commuting pair, 40Companion matrix, 265Complement, 75

orthogonal, 374Completely reducible, 49Complex conjugate, 8Complex Hadamard matrix, 388Complex number, 7Complex polarization identity, 363Complexification, 72Condensation,

method of, 230Condition number, 210

spectral, 432Congruent matrices, 457Conjugate,

complex, 8Conjugate transpose, 334Continuant, 243Convex subset, 412Convolution algebra, 41Copositive, 418Coproduct

direct, 26Courant–Fischer Minimax Theorem, 400Cramer’s Theorem, 236Cross product, 42Crout’s algorithm, 169Cut norm, 345Cyclic, 117Cyclotomic polynomial, 48

DDecomposition

Cholesky, 406direct sum, 74LU, 168polar, 431QR, 379singular value, 431spectral, 429

Defective eigenvalue, 270Deflation, 281Degree

algebraic, 110of a generalized eigenvector, 304of a multivariate polynomial, 50of a polynomial, 44of a polynomial function, 47

Derivation, 114Derogatory endomorphism, 270Determinant, 227Determinant function, 221Diagonal Dominance Theorem, 350Diagonal matrix, 148Diagonalizable endomorphism, 261Difference set, 2Differential, 90Differential operator, 114Dimension, 71Dirac functional, 321Direct coproduct, 26Direct product, 23Direct sum, 74Direct sum decomposition, 74Discrete cosine transform, 152Discrete Fourier transform, 152, 341Disjoint subspaces, 27Distance, 352Distance matrix, 400Distribution, 318Division algebra, 52Division Algorithm, 46Domain, 2

integral, 9Dominant eigenvalue, 280Dot product, 334

Index 491

weighted, 335Drazin pseudoinverse, 450Dual basis, 323Dual space, 317

weak, 323Dyadic product, 145

EEigenspace, 257, 258

generalized, 306Eigenvalue, 255, 258

defective, 270dominant, 280semisimple, 270simple, 270

Eigenvector, 255, 258generalized, 304

Eisenstein’s criterion, 47Elementary automorphism, 115Elementary matrix, 153Elementary operation, 158Elliptic norm, 403Endomorphism, 113

almost positive semidefinite, 415bounded, 125copositive, 418derogatory, 270diagonalizable, 261indefinite, 403integration, 114nilpotent, 302nonderogatory, 270normal, 424orthogonally diagonalizable, 398positive definite, 403positive semidefinite, 403selfadjoint, 395

Entangled tensors, 465Entire, 44Epic, 2Epimorphism, 95Equal functions, 2Equivalence relation, 120Equivalent matrices, 161Equivalent norms, 346Euclidean

norm, 338subfield, 333

Evaluation functional, 324Even function, 76Even permutation, 224Exchange Property, 29

Extended coefficient matrix, 191Extension, 2Exterior algebra, 470Exterior power, 470Exterior product, 136Exterior square, 470

FFactor space, 110Fan, 186Fast Fourier transform, 152Fibonacci sequence, 299Field, 5

algebraically closed, 49formally-real, 334Galois, 8orderable, 18

Field of algebraic numbers, 73Finite dimensional, 71Finitely generated, 29Fixed point, 256Fixed space, 257Flat polynomial, 51Form

bilinear, 455Lorentz, 462quadratic, 461

Formal differentiation, 117Formal Laurent series, 18Formally real field, 334Fourier coefficient, 378Fredholm alternative, 326Frobenius norm, 345Full pivoting, 168Function, 2

characteristic, 24determinant, 221even, 76inverse, 2linearly independent, 110odd, 76periodic, 69piecewise constant, 34spline, 56weight, 318

FunctionalDirac, 321evaluation, 324linear, 317zero, 317

Fundamental Theorem of Algebra, 49Fuzzification, 87Fuzzy subspace, 37

492 Index

GGalois field, 8Gap, 387Gauss–Jordan method, 195Gauss–Seidel iteration method, 205Gaussian elimination, 195Gelfand–Naimark Property, 449General Lie algebra, 147General quadratic equation, 464Generalized characteristic polynomial, 282Generalized eigenspace, 306Generalized eigenvector, 304Generating set, 28Geometric multiplicity, 270Gershgorin bound, 349Gershgorin’s Theorem, 350Givens rotation matrix, 420GMRES algorithm, 375Golden ratio, 299Google matrix, 282Gram matrix, 336Gram–Schmidt process, 372Gram–Schmidt Theorem, 372Graph, 91Grassmann’s Theorem, 77Greville’s method, 444Grothendieck’s inequality, 345Group of automorphisms, 120Guttman’s Theorem, 154

HHaar wavelet, 376Hadamard inequality, 373Hadamard matrix, 229

complex, 388Hadamard product, 174Hahn–Banach Theorem, 347Hamel basis, 69Hamming norm, 352Hankel matrix, 241Hausdorff Maximum Principle, 67Hermitian matrix, 396Hermitian transpose, 334Hibert subset, 376Hilbert matrix, 159Hilbert–Schmidt norm, 345Homogeneous system of linear equations, 190Homomorphism, 89

of algebras, 89of unital algebras, 89

Householder matrix, 423Hua’s identity, 12

Hulluniform, 38zero-sum, 38

Hyperplane, 325

IIdempotent, 118Ill-conditioned, 209Image, 94Imaginary part, 7Improper subspace, 25Indefinite endomorphism, 403Independent subspaces, 74Indeterminate, 44Index of nilpotence, 302Induced norm, 343Infinite dimensional, 71Inner product, 333Inner product space, 333Integral domain, 9Integration endomorphism, 114Interior product, 136Interpolation problem, 189Intersection, 1Invariant subspace, 117Inverse function, 2Inversion, 224Involution, 385Involutory matrix, 151Irreducible, 47Isometry, 409Isomorphic vector spaces, 97Isomorphism, 95

of algebras, 95of unital algebras, 95

Iteration methodGauss–Seidel, 205Jacobi, 205JOR, 207SOR, 207stationary, 208

JJacobi identity, 41Jacobi iteration method, 205Jacobi overrelaxation method, 207Jacobi polynomial, 371Jacobi reflection matrix, 422JOR, 207Jordan algebra, 43Jordan canonical form, 304, 308Jordan identity, 43Jordan product, 43

Index 493

KKaratsuba’s algorithm, 45Kernel, 94Ket-bra product, 136Kovarik algorithm, 445Kronecker product, 467Krylov algorithm, 301Krylov subspace, 297

LLagrange identity, 339Lagrange interpolation polynomial, 162Lanczos algorithm, 301Laurent series, formal, 18Leading coefficient, 44Leading entry, 194Least squares method, 447Legendre polynomial, 370Lie algebra, 41

general, 147special, 320

Lie product, 42Lightcone inequality, 463Linear combination, 27Linear functional, 317Linear transformation, 89

adjoint of, 383Linear variety, 96Linearly dependent, 57

locally, 92Linearly independent, 57, 110

nearly, 82with respect to a fuzzification, 87

Linearly recurrent sequence, 298List, 1Locally linearly dependent, 92Loewner partial order, 404Lorentz form, 462Lower-triangular matrix, 150LU-decomposition, 168

MMagic matrix, 287Markov matrix, 150Matrices

column-equivalent, 161congruent, 457equivalent, 161Pauli, 82row-equivalent, 161similar, 266unitarily similar, 420

Matrix, 23adjacency, 399

adjoint, 235anti-Hermitian, 413band, 149change-of-basis, 162circulant, 172coefficient, 191companion, 265complex Hadamard, 388determinant of, 227diagonal, 148distance, 400elementary, 153extended coefficient, 191Givens rotation, 420Google, 282Gram, 336Hadamard, 229Hankel, 241Hermitian, 396Hilbert, 159Householder, 423in block form, 138in reduced row echelon form, 194in row echelon form, 193involutory, 151Jacobi reflection, 422lower-triangular, 150magic, 287Markov, 150Nievergelt’s, 159nonsingular, 151normal, 434orthogonal, 422permutation, 158quasidefinite, 415scalar, 148singular, 151skew-symmetric, 150sparse, 167special orthogonal, 424stochastic, 150strictly diagonally dominant, 249symmetric, 150symmetric Toeplitz, 202symplectic, 440transpose, 95tridiagonal, 149unitary, 419upper Hessenberg, 473upper-triangular, 149Vandermonde, 163zero, 24

Matrix logarithm, 356Matrix representation, 153

494 Index

Maximal, 61Maximal subspace, 325Method of condensation, 230Minimal, 61Minimal polynomial, 273, 298Minkowski’s inequality, 339Minor, 230Modular Law, 31Moebius function, 48Monic

function, 2polynomial, 44

Monomorphism, 95Moore–Penrose pseudoinverse, 441Multiplication

in a field, 5scalar, 21

Multiplication table, 65Multiplicity

algebraic, 270geometric, 270

Mutually orthogonal, 369

NNearly linearly independent, 82Nevanlinna–Pick Interpolation Theorem, 403Nievergelt’s matrix, 159Nilpotent, 302Nondegenerate bilinear form, 455Nonderogatory endomorphism, 270Nonhomogeneous system of linear equations,

190Nonsingular matrix, 151Nontrivial subspace, 25Norm, 338, 342

cut, 345elliptic, 403Euclidean, 338Frobenius, 345Hamming, 352Hilbert–Schmidt, 345induced, 343spectral, 344triangular, 37

Normal endomorphism, 424Normal matrix, 434Normal vector, 338Normed space, 342Norms

equivalent, 346Nullity, 98Number

complex, 7rational, 5

real, 5Numerical range, 463

OOdd function, 76Odd permutation, 224Operation

elementary, 158Opposite transformation, 453Optimization algebra, 9Order of recurrence, 298Orderable field, 18Orthogonal, 369Orthogonal complement, 374

right, 458Orthogonal matrix, 422Orthogonal projection, 374Orthogonality with respect to a bilinear form,

458Orthogonally diagonalizable, 398Orthonormal, 375

PPadé approximant, 239Pairwise disjoint, 27Parallelogram law, 339Parity, 456Parseval’s identity, 380Partial order, 60

Loewner, 404Partial pivoting, 168Partially-ordered set, 60Pauli matrices, 82Periodic function, 69Permanent, 252Permutation, 2

even, 224odd, 224

Permutation matrix, 158Pfaffian, 228Piecewise constant, 34Pivot, 168Pivoting

full, 168partial, 168

Poincaré–Birkhoff–Witt Theorem, 42Polar decomposition, 431Polarization identity

complex, 363real, 363

Polynomial, 44characteristic, 264, 298Chebyshev, 370completely reducible, 49

Index 495

Polynomial (cont.)cyclotomic, 48flat, 51in several indeterminates, 50irreducible, 47Jacobi, 371Lagrange interpolation, 162Legendre, 370minimal, 273, 298monic, 44reciprocal, 440reducible, 47trigonometric, 56zero, 44

Polynomial function, 47Positive definite, 403Positive quadratic form, 462Positive semidefinite, 403Power

exterior, 470Pre-Banach space, 342Pre-Hilbert space, 333Primitive root of unity, 152Principal component analysis, 118Process

Arnoldi, 375Gram–Schmidt, 372

Productbra-ket, 136Cartesian, 2cross, 42direct, 23dot, 334dyadic, 145exterior, 136Hadamard, 174inner, 333interior, 136Jordan, 43ket-bra, 136Kronecker, 467Lie, 42scalar triple, 339Schur, 174tensor, 465vector triple, 339

Projection, 118onto an affine set, 387orthogonal, 374

Proper subspace, 25Pseudoinverse

Drazin, 450Moore–Penrose, 441

QQR algorithm, 380QR-decomposition, 379Quadratic form, 461

positive, 462Quadratic surface, 464Quasidefinite matrix, 415Quaternion

algebra, 66real, 66

QZ algorithm, 380

RRange, 2

numerical, 463Rank, 98, 199, 457Rational Decomposition Theorem, 304Rational number, 5Rayleigh quotient

function, 400iteration scheme, 400

Real Euclidean, 333Real number, 5Real part, 7Real polarization identity, 363Real quaternion, 66Reciprocal polynomial, 440Reduced row echelon form, 194Reducible, 47Relation

equivalence, 120partial order, 60

Relaxation method, 207Representation, 153Restriction, 2Riesz Representation Theorem, 382Right orthogonal complement, 458Row echelon form, 193Row equivalent, 161Row space, 199

SScalar, 22Scalar matrix, 148Scalar multiplication, 21Scalar triple product, 339Schur complement, 160Schur product, 174Schur’s Theorem, 421Selfadjoint, 395Semifield, 9Semisimple eigenvalue, 270

496 Index

Sequence, 1Fibonacci, 299linearly recurrent, 298

Setdifference, 2generating, 28partially-ordered, 60spanning, 28

Sherman–Morrison–Woodbury Theorem, 154Signum, 224Similar matrices, 266Simple eigenvalue, 270Simple tensors, 465Simpson’s rule, 187Singular matrix, 151Singular value, 432Singular Value Decomposition Theorem, 431Skew symmetric bilinear form, 453Skew symmetric matrix, 150Solution set, 191Solution space, 191SOR, 207Space

dual, 317inner product, 333normed, 342pre-Banach, 342pre-Hilbert, 333solution, 191

Spanning set, 28Sparse matrix, 167Special Lie algebra, 320Special orthogonal matrix, 424Spectral condition number, 432Spectral Decomposition Theorem, 429Spectral norm, 344Spectral radius, 258Spectrum, 255Spline function, 56Stabilizer, 131Standard identity, 238Stationary iteration method, 208Steinitz Replacement Property, 60Stochastic matrix, 150Strassen–Winograd algorithm, 166Strictly diagonally dominant, 249Subalgebra, 41

unital, 41Subfield, 6

Euclidean, 333real Euclidean, 333

Subsetaffine, 96bounded, 67

chain, 61convex, 412Hilbert, 376orthonormal, 375underlying, 1

Subspace, 25cyclic, 117fuzzy, 37generated by, 28improper, 25invariant, 117Krylov, 297maximal, 325nontrivial, 25proper, 25spanned by, 28trivial, 25

Subspacesdisjoint, 27independent, 74pairwise disjoint, 27

Successive overrelaxation method, 207Sylvester’s Theorem, 98Symmetric

bilinear transformation, 453matrix, 150Toeplitz matrix, 202with respect to an involution, 385

Symmetric difference, 24Symplectic matrix, 440System of linear equations, 189

homogeneous, 190nonhomogeneous, 190

TTaber’s Theorem, 321Taylor coefficient, 117Tensor algebra, 470Tensor product, 465Tensors

entangled, 465simple, 465

Ternary ring, 18Trace, 318Transcendental, 73Transform

discrete cosine, 152discrete Fourier, 152, 341fast Fourier, 152

Transformationaffine, 96bilinear, 453linear, 89opposite, 453

Index 497

Transpose, 95conjugate, 334Hermitian, 334

Triangle difference inequality, 339Triangle inequality, 353Triangular norm, 37Tridiagonal matrix, 149Trigonometric polynomial, 56Trivial subspace, 25

UUnderlying subset, 1Uniform combination, 38Uniform hull, 38Union, 1Unit, 40Unital algebra, 39Unital subalgebra, 41Unitarily similar matrices, 420Unitary automorphism, 419Unitary matrix, 419Upper Hessenberg matrix, 473Upper-triangular matrix, 149

VVandermonde matrix, 163Variety

linear, 96Vector, 22

normal, 338

Vector addition, 21Vector space, 21

finite dimensional, 71finitely generated, 29infinite dimensional, 71

Vector triple product, 339Vectors

orthogonal, 369orthogonal with respect to a bilinear form,

458

WWavelet

Haar, 376Weak dual space, 323Weight function, 318Weight of a Baxter algebra, 90Weighted dot product, 335Well Ordering Principle, 61Weyl’s Problem, 399Width of a band matrix, 149Wiedemann algorithm, 301Word, 23Wronskian, 228

ZZero-sum combination, 38Zero-sum hull, 38Zlobec’s formula, 445Zorn’s Lemma, 67