Selected Titles in This Series
151 S . Yu . Slavyanov , Asymptotic solutions o f the one-dimensional Schrodinge r equation , 199 6 150 B . Ya. Levin , Lectures on entire functions, 199 6 149 Takash i Sakai, Riemannian geometry , 199 6 148 Vladimi r I. Piterbarg, Asymptotic methods i n the theory o f Gaussian processe s and fields, 1996 147 S . G . Gindiki n an d L. R. Volevich , Mixed problem fo r partia l differentia l equation s with quasihomogeneous principa l part, 199 6 146 L . Ya. Adrianova , Introduction t o linear system s of differential equations , 199 5 145 A . N. Andrianov and V. G. Zhuravlev , Modular form s an d Heck e operators, 199 5 144 O . V. Troshkin, Nontraditional method s in mathematical hydrodynamics , 199 5 143 V . A. Malyshev an d R. A. Minlos, Linear infinite-particl e operators , 199 5 142 N . V . Krylov, Introduction t o the theory o f diffusio n processes , 199 5 141 A . A. Davydov, Qualitative theor y o f control systems , 199 4 140 Aizi k I. Volpert, Vitaly A. Volpert, an d Vladimir A. Volpert, Traveling wav e solutions o f parabolic systems, 199 4 139 I . V . Skrypnik, Methods for analysi s o f nonlinear ellipti c boundary valu e problems, 199 4 138 Yu . P. Razmyslov, Identities o f algebras and their representations, 199 4 137 F . I. Karpelevich and A. Ya. Kreinin , Heavy traffic limit s for multiphas e queues, 199 4 136 Masayosh i Miyanishi, Algebraic geometry, 199 4 135 Masar u Takeuchi, Modern spherica l functions, 199 4 134 V . V. Prasolov, Problems and theorem s in linear algebra, 199 4 133 P . I. Naumkin an d I. A. Shishmarev, Nonlinear nonloca l equations in the theory o f waves, 199 4 132 Hajim e Urakawa, Calculus of variations an d harmonic maps, 199 3 131 V . V. Sharko, Functions on manifolds: Algebrai c and topological aspects, 199 3 130 V . V. Vershinin, Cobordisms and spectral sequences , 199 3 129 Mitsu o Morimoto, An introduction t o Sato' s hyperfunctions, 199 3 128 V . P. Orevkov, Complexity o f proofs an d their transformations i n axiomatic theories, 199 3 127 F . L. Zak, Tangents and secants o f algebraic varieties, 199 3 126 M . L. Agranovskii, Invariant functio n space s o n homogeneous manifold s o f Lie groups and applications, 199 3 125 Masayosh i Nagata, Theory o f commutative fields, 1993 124 Masahis a Adachi, Embeddings an d immersions, 199 3 123 M . A. Akivis and B. A. Rosenfeld, Eli e Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hou, Theory o f entire and meromorphic functions : Deficien t an d asymptoti c value s and singula r directions, 199 3 121 LB . Fesenk o an d S. V . Vostokov, Local fields and thei r extensions: A constructive approach, 199 3 120 Takeyuk i Hid a an d Masuyuki Hitsuda, Gaussian processes , 199 3 119 M . V . Karasev an d V. P. Maslov, Nonlinear Poisso n brackets. Geometry an d quantization, 199 3 118 Kenkich i Iwasawa, Algebraic functions, 199 3 117 Bori s Zilber, Uncountably categorica l theories , 199 3 116 G . M. Fel'dman, Arithmetic of probability distributions , and characterization problem s on abelian groups, 199 3 115 Nikola i V . Ivanov, Subgroups o f Teichmuller modular groups , 199 2 114 Seiz o ltd, Diffusion equations , 199 2 113 Michai l Zhitomirskil , Typical singularities o f differential 1-form s and Pfaffia n equations , 199 2 112 S . A. Lomov, Introduction t o the general theory o f singular perturbations, 199 2 111 Simo n Gindikin, Tube domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introduction t o comple x analysis Part II. Functions o f severa l variables, 199 2
(Continued in the back of this publication) Problems and Theorems in Linear Algebra This page intentionally left blank V.V. Prasoiov Problems and Theorems in Linear Algebra 10.1090/mmono/134
BHKTOP BacHJibeBir a IIpacojiO B 3AHAMH H TEOPEML I JIHHEHHOH AJirEBPb l
Translated b y D . A . Leite s fro m a n origina l Russia n manuscrip t Translation edite d b y Simeo n Ivano v
2000 Mathematics Subject Classification. Primar y 15-01 .
ABSTRACT. Thi s boo k contain s the basic s o f linear algebr a with a n emphasi s o n nonstandard an d neat proof s o f known theorems. Man y o f the theorems o f linear algebra obtained mainly during th e past thirt y year s ar e usuall y ignore d i n textbooks bu t ar e quit e accessibl e fo r student s majorin g or minorin g i n mathematics . Thes e theorem s ar e give n wit h complet e proofs . Ther e ar e abou t 230 problems wit h solutions .
Library o f Congres s Cataloging-in-Publicatio n Dat a Prasolov, V . V . (Vikto r Vasil'evich ) [Zadachi i teoremy linemo i algebry . English ] Problems an d theorem s i n linea r algebra/V . V . Prasolov ; [translate d b y D . A . Leite s fro m a n original Russia n manuscript ; translatio n edite d b y Simeo n Ivanov ] p. cm . — (Translation s o f mathematical monographs , ISS N 0065-9282 ; v. 134 ) Includes bibliographica l references . ISBN 0-8218-0236- 4 1. Algebras , Linear . I . Ivanov , Simeon . II . Title . III . Series .
QA184.P7313 199 4 94-1333 2 512/.5—dc20 CI P
Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , or multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed to the Acquisitions Department, America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-229 4 USA . Request s ca n als o b e mad e b y e-mail t o [email protected] .
© 199 4 b y the America n Mathematica l Society . Al l rights reserved . Reprinted wit h correction s 1996 . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . & Printe d o n recycle d paper . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 1 3 1 2 1 1 1 0 0 9 0 8 CONTENTS
Preface x v Main notations and conventions xvi i Chapter I. Determinant s 1 Historical remarks: Leibni z and Sek i Kova. Cramer , L'Hospital, Cauch y and Jacobi 1. Basic properties of determinants 1 The Vandermonde determinant an d its application. Th e Cauchy determinant. Continue d frac - tions and the determinant of a tridiagonal matrix. Certai n other determinants. Problems 2. Minors and cofactors 9 Binet-Cauchy's formula. Laplace' s theorem. Jacobi' s theorem on minors of the adjoint matrix. The generalize d Sylvester' s identity . Chebotarev' s theore m o n th e matri x ||£ u||f~ , wher e e — exp(2ni/p). Problems 3. The Schur complement 1 6 GivenA = ( n 1 2 ),thematrix(^4|^ii ) = A22—A21A7, 1 A\2 is called the Schur complement \A2\ A22J (of A\\ in A).
3.1. de t A = de t A u de t (A\A n). 3.2. Theorem . (A\B) = ((A\C)\(B\C)). Problems 4. Symmetri c functions, sums x\ H f-xf , an d BernoulU numbers 1 9
Determinant relations between ok (x\,..., x n), J * (X\ ,..., x„ ) = x f H hx j and p^ (x\, ..., x„ ) = l l n ^2 x\ ... x n . A determinant formula for S„ (k) = 1 " -\ h (k — 1)". The Bernoulli numbers and S n(k). 4.4. Theorem . Let u = S\(x) andv — S2W. Then for k > 1 there exist polynomials p^ and q^ such that S2k+\ W = u 2pjc(u) and S^M = v 4k(u)- Problems Solutions
Chapter II. Linea r spaces 3 5 Historical remarks: Hamilton and Grassmann 5. The dual space. The orthogonal complement 3 7 Linear equations and their application to the following theorem: Vlll CONTENTS
5.4.3. Theorem . If a rectangle with sides a andb is arbitrarily cut into squares with sides x\,... ,x„ then — € Q and — £ Qfor all i. a b Problems 6. Th e kernel (null space) and the image (range) o f an operator. Th e quotient space 6.2.1. Theorem . Ker^* = (ImA) 1- andlmA* = (KQTA^. Fredholm's alternative . Kronecker-Capelli' s theorem . Criteri a fo r solvabilit y o f th e matri x equation C = AXB. Problem 7. Bases of a vector space. Linear independence Change of basis. Th e characteristic polynomial.
7.2. Theorem . Let x\ ,..., x„ and y\,...,y n be two bases, 1 < k < n. Then k of the vectors y\,- • • >yn can be interchanged with some k of the vectors x\,..., x„ so that we get again two bases. 7.3. Theorem . Let T : V — • V be a linear operator such that the vectors £, T?,..., T n£, are linearly dependent for every £ £ V. Then the operators I, T,..., T" are linearly dependent. Problems 8. The rank of a matrix The Frobenius inequality. Th e Sylvester inequality.
8.3. Theorem . Let U be a linear subspace of the space Mn,m ofnxm matrices, and r < m < n. If rank X < r for any X G U then dim U < rn.
A description of subspaces U C M„, m such that dim U = nr. Problems 9. Subspaces. The Gram-Schmidt orthogonalization process Orthogonal projections. 9.5. Theorem . Let e\,...,e„ be an orthogonal basis for a space V, d\ — ||e/1|. The projections of the vectors e\,...,e n onto an m-dimensional subspace of V have equal lengths if and only if d}{d~2 + h d~2) > mfor every i = 1,... , n. 9.6.1. Theorem . A set of k-dimensional subspaces of V is such that any two of these subspaces have a common (k — 1 )-dimensional subspace. Then either all these subspaces have a common {k — 1)- dimensional subspace or all of them are contained in the same {k H - I)-dimensional subspace. Problems 10. Complexification and realisation. Unitar y spaces Unitary operators. Norma l operators. 10.3.4. Theorem . Let B and C be Hermitian operators. Then the operator A = B + iC is normal if and only ifBC = CB. Complex structures. Problems Solutions Chapter 1H. Canonica l forms of matrices and linear operators 11. The trace and eigenvalues of an operator The eigenvalue s o f a n Hermitia n operato r an d o f a unitar y operator . Th e eigenvalue s o f a tridiagonal matrix. Problems 12. The Jordan canonical (normal) for m 12.1. Theorem . If A and B are matrices with real entries and A = PBP~ l for some matrix P with complex entries then A = QBQ~ ] for some matrix Q with real entries. The existence and uniqueness of the Jordan canonical for m (Valiacho's simple proof). CONTENTS IX
The real Jordan canonical form. 12.5.1. Theorem , a ) For any operator A there exist a nilpotent operator A n and a semisimple operator As such that A = A s - f An and AsAn = A nAs. b) The operators An andA s are unique; besides, As = S(A) andA n = N (A) for some polynomials SandN. 12.5.2. Theorem. For any invertible operator A there exist a unipotent operator Au and a semisimple operator As such that A — ASAU = AUAS. Such a representation is unique. 12.6. The proof of the Kronecker theorem for points of linear maps. Problems 13. The minimal polynomial and the characteristic polynomial 7 7 13.1.2. Theorem. For any operator A there exists a vector v such that the minimal polynomial ofv {with respect to A) coincides with the minimal polynomial of A. 13.3. Theorem. The characteristic polynomial of a matrix A coincides with its minimal polynomial k if and only if for any vector (x\9...txn) there exist a column P and a row Q such that x* = QA P. Hamilton-Cayley's theorem and its generalization for polynomials of matrices. Problems 14. The Frobenius canonical form 8 0 Existence of the Frobenius canonical form (H. G. Jacob's simple proof) Problems 15. How to reduce the diagonal to a convenient form 8 1 15.1. Theorem. If A ^ XI then A is similar to a matrix with the diagonal elements (0,..., 0, tr A). 15.2. Theorem. Any matrix A is similar to a matrix with equal diagonal elements. 153. Theorem . Any nonzero square matrix A is similar to a matrix all diagonal elements of which are nonzero. Problems 16. The polar decomposition 8 4 The polar decomposition of noninvertible and of invertible matrices. The uniqueness of the polar decomposition of an invertible matrix. 16.1. Theorem . IfA — S\U\ = U2S2 are polar decompositions of an invertible matrix A then Vx = U 2. 16.2.1. Theorem . For any matrix A there exist unitary matrices U, W and a diagonal matrix D suchthatA = UDW. Problems 17. Factorizations of matrices 8 6 17.1. Theorem. For any complex matrix A there exist a unitary matrix U and a triangular matrix T such that A = UTU*. The matrix A is a normal one if and only if T is a diagonal one. Gauss, Gram, and Lanczos factorizations. 17.3. Theorem. Any matrix is a product of two symmetric matrices. Problems 18. The Smith normal form. Elementar y factors of matrices 8 7 Problems Solutions Chapter IV. Matrice s of special form 9 5 19. Symmetric and Hermitian matrices 9 5 Sylvester's criterion. Sylvester's law of inertia. Lagrange's theorem on quadratic forms. Courant - Fisher's theorem. 19.5.1.Theorem. If A >0and(Ax,x) = Ofor any x, then A = 0. Problems X CONTENTS
20. Simultaneous diagonalization of a pair of Hermitian forms Simultaneous diagonalization of two Hermitian matrices A and B when A > 0 . A n example of two Hermitian matrices which can not be simultaneously diagonalized . Simultaneou s diagonaliza - tion of two semidefinite matrices . Simultaneou s diagonalization o f two Hermitian matrices A an d B such that there is no x ^ 0 for which x*Ax — x*Bx = 0 . Problems 21. Skew-symmetri c matrices 21.1.1. Theorem . If A is a skew-symmetric matrix then A2 < 0 . 21.1.2. Theorem . If A is a real matrix such that (Ax, x) — Ofor all x, then A is a skew-symmetric matrix. 21.2. Theorem . Any skew-symmetric bilinear form can be expressed as
r
k=\ Problems 22. Orthogonal matrices. The Cayley transformation The standar d Cayle y transformatio n o f a n orthogona l matri x whic h doe s no t hav e 1 as it s eigenvalue. Th e generalize d Cayle y transformatio n o f a n orthogona l matri x whic h ha s 1 as it s eigenvalue. Problems 23. Normal matrices 23.1.1. Theorem . If an operator A is normal then Ker A* = Ke r A and Im A* = ImA. 23.1.2. Theorem . An operator A is normal if and only if any eigenvector of A is an eigenvector of A*. 23.2. Theorem . If an operator A is normal then there exists a polynomial P such that A* = P(A). Problems 24. Nilpotent matrices 24.2.1. Theorem . Let A be an n x n matrix. The matrix A is nilpotent if and only iftv(A p) = 0 for each p = 1,... , n. Nilpotent matrices and Young tableaux. Problems 25. Projections. Idempoten t matrices 25.2.1&2. Theorem . An idempotent operator P is an Hermitian one if and only if a) Ker P _ L Im P; orb) \Px\ < \x\ for every x.
25.2.3. Theorem . Let P\,...,P n be Hermitian, idempotent operators. The operator P = P\ + • • • + P n is an idempotent one if and only ifPiPj = 0 whenever i ^ j. 25.4.1. Theorem . Let V\ ® • • • © Vk,Pi '- V — • V( be Hermitian idempotent operators,
A = Pi H + P k. Then 0 < de t A < 1 and det A = 1 if and only ifVt J _ Vj whenever i ^ / . Problems 26. Involutions 26.2. Theorem . A matrix A can be represented as the product of two involutions if and only if the matrices A and A~l are similar. Problems Solutions Chapter V. Multilinea r algebra 27. Multilinear maps and tensor products 119 CONTENTS XI
An invariant definition o f the trace. Kronecker's product of matrices, A<8 > B; the eigenvalues of the matrices A®B an d A ® / + / ® B. Matri x equations AX - XB = C and AX - XB = XX. Problems 28. Symmetri c and skew-symmetric tensors The Grassmann algebra. Certain canonical isomorphisms. Applications of Grassmann algebra: proofs of Binet-Cauchy's formula and Sylvesters identity. q 28.5.4. Theorem . Let AB(t) = 1 + £ tr(A|)f * andS B(t) = 1 + £ tr(S B)t^ Then SB(t) =
(AB(-t))~K Problems 29. ThePfaffia n +j,+1 The Pfaffian o f principal submatrices of the matrix M = \\mtj || j", wher e m,7 = (—l)' . 29.2.2. Theorem . Given a skew-symmetric matrix A we have
A PfU + *M) = £ X* Pk, where Pk = Y, h " ' "^ ) • k=0 o \ l '" 2(n-k) J
Problems 30. Decomposable skew-symmetric and symmetric tensors 30.2.1. Theorem . x\ A • • • A xk = y\ A • • • A y k ^ 0 if and only if Span(xi,... , xk) = Span (>>i,..., j^). 30.2.2. Theorem . S{x\ 0 • • •
A complete description of the following types of transformations of
m n V ®{V*) ^Mm,n:
1) rank-preserving; 2) determinant-preserving; 3) eigenvalue-preserving; 4) invertibility-preserving. Problems Solutions Chapter VI. Matri x inequalities 33. Inequalities for symmetric and Hermitian matrices 33.1.1. Theorem . IfA>B>0 then A~l < B~ l. 33.1.3. Theorem . If A > 0 isa real matrix then
[A~lx, x) = max(2{x,y) — (Ay, y)). y Xll CONTENTS
B 33.2.1. Theorem . Suppose A=(*\ J > 0 . Then \A\ < \A\ | • \A2\. Hadamard's inequality and Szasz' s inequality. n 33.3.1. Theorem . Suppose a, > 0 , £3 a,- = 1 and Aj > 0 . Then i=\
\aiAi + • • • + a kAk\ > Mip + • • • + \A k\°*.
33.3.2. Theorem . Suppose Aj > 0 , a,- G C 77*e «
| det(oMi + • • • + a kAk)\ < det(|a i M, + • • • + \a k\Ak).
Problems 34. Inequalitie s for eigenvalue s Schur's inequality. Weyl' s inequality (fo r eigenvalues of A + B). 34.2.2. Theorem . Let A = ( ) > 0 be an Hermitian matrix, a\ < • • • < a „ ^«fl ? \C* Bj /^l < — • < Pm the eigenvalues of A and B, respectively. Then a; < ft < a„ +;_m. 34.3. Theorem . Le/ ^ 4 and B be Hermitian idempotents, X any eigenvalue of AB. Then 0 < X < 1 . 34.4.1. Theorem . Let the kf and pi be the eigenvalues of A and A A*, respectively; let o\ = y/jij.
Let \k\ < • • • < k„, where n is the order of A. Then \k\ ... k m \
The spectral norm H^l ^ an d the Euclidean norm \\A\\ e, th e spectral radius p(A). 35.1.2. Theorem . If a matrix A is normal then p(A) = \\A\\ S. 35.2. Theorem . \\A\\, < \\A\\, < rf\\A\\,. The invariance of the matrix norm and singular values. A + A* 35.3.1. Theorem . Let S be an Hermitian matrix. Then \\A 1 | does not exceed \\A — S\\, where ||-|| is the Euclidean or operator norm. 35.3.2. Theorem . Let A = US be the polar decomposition of A and W a unitary matrix. Then
IM-^||e<||^-^||e andif\A\ ^ 0 , then the equality is only attained for W = U. Problems 36. Schur' s complement and Hadamard's product. Theorems of Emily Haynsworth
36.1.1. Theorem . IfA>0 then (A\A u) > 0. 36.1.4. Theorem . IfA k andB k are the kth principal submatrices of positive definite order n matrices A and B, then M + ^w(ltgj4})tW(1+gg}).
Hadamard's product A o B. 36.2.1. Theorem . IfA>0andB>0 then A o B > 0. Oppenheim's inequalit y Problems 37. Nonnegativ e matrices Wielandt's theore m Problems CONTENTS xiu
38. Doubly stochastic matrices Birkhoff's theorem . H.Weyl' s inequality. Solutions Chapter VII. Matrice s in algebra and calculus 39. Commuting matrices The space of solutions of the equation AX — XA fo r X wit h the given A o f order n. 39.2.2. Theorem . Any set of commuting diagonalizable operators has a common eigenbasis. 39.3. Theorem . LetA,B be matrices such that AX = XA implies BX = XB. Then B = g(A), where g is a polynomial. Problems 40. Commutator s 40.2. Theorem . If tr A = 0 then there exist matrices X and Y such that [X, Y] = A and either (1 ) tr Y — 0 and an Hermitian matrix X or (2) X and Y have prescribed eigenvalues. 40.3. Theorem . Let A,B be matrices such that ad ^ X = 0 implies ad^ B = Ofor some s > 0 . Then B = g(A)for a polynomial g.
40.4. Theorem . Matrices A\,...,A n can be simultaneously triangularized over C if and only if the matrix p{A\, ..., A n)[Aj,Aj] is a nilpotent one for any polynomial p(x\, ..., x n) in noncommuting indeterminates. 40.5. Theorem . If ra.nk[A, B] < 1 , then A and B can be simultaneously triangularized over C. Problems 41. Quaternion s and Cayley numbers. Cliffor d algebra s Isomorphisms so(3, R) *£ su(2) and so(4, R) *£ so(3, R) 0 so(3 , R). Th e vector products in R3 and R7. Hurwitz-Rado n familie s of matrices. Hurwitz-Radon ' number p(2c+4d (2a + 1) ) = 2 C + Sd. 41.7.1. Theorem . The identity of the form
2 (X\ + ... + X l){y\ + ... +yl) = (Z\ + ... + Z„ ), where Zj(x, y) is a bilinear function, holds if and only ifm < p{n). 41.7.5. Theorem . In the space of real n x n matrices, a subspace ofinvertible matrices of dimension m exists if and only ifm < p(n). Other applications: algebra s with norm, vector product, linear vector fields on spheres. Clifford algebra s and Clifford modules . Problems 42. Representation s of matrix algebras Complete reducibility of finite-dimensional representation s o f Mat(F"). Problems 43. The resultant Sylvester's matrix, Bezout's matrix and Barnett's matrix Problems 44. The general inverse matrix. Matri x equations 44.3. Theorem , a ) The equation AX — XA = C is solvable if and only if the matrices I 1 and ( ) are similar. O B J b) The equation AX — YA — C is solvable if and only (/"rank ( ] = ran k ( Problems 45. Hanke l matrices and rational function s 200 XIV CONTENTS
46. Functions of matrices. Differentiation o f matrices Differential equatio n X = AX an d the Jacobi formula for det A. Problems 47. Lax pairs and integrable systems 48. Matrices with prescribed eigenvalues n n ] 48.1.2. Theorem . For any polynomial f(x) = x + c\x ~ -\ \- cn and any matrix B of order n — 1 whose characteristic and minimal polynomials coincide there exists a matrix A such that B is a submatrix of A and the characteristic polynomial of A is equal to f. 48.2. Theorem . Given all offdiagonal elements in a complex matrix A it is possible to select diagonal elements x\,...,x„ so that the eigenvalues of A are given complex numbers; there are finitely many sets {xi,..., x n} satisfying this condition. Solutions Appendix Eisenstein's criterion, Hilbert's Nullstellensatz. Bibliography Subject Index PREFACE
There are very many books on linear algebra, among them many really wonderfu l ones (see the list of recommended literature). One might think that no more books on this subject are necessary. Choosing the words more carefully, it is possible to deduce that these books contain all that one needs and in the best possible form, and therefore any new book will, at best, only repeat the old ones. This opinion is manifestly wrong, but nevertheless almost ubiquitous. New results in linear algebra appear constantly and so do new, simpler and neater proofs o f the known theorems. Besides , more than a few interesting old results are ignored by textbooks. In this book I tried to collect the most attractive problems and theorems of linear algebra still accessible to students majoring in mathematics. The computational aspects of linear algebra were left somewhat aside. Th e major part of the book contains results known from journal publications only. I believe that those results will be of interest to many readers. I assume that the reader is acquainted with the main notions of linear algebra: linear space, basis, hnear map, and the determinant o f a matrix. Apar t fro m this , all the essential theorems of the standard course of linear algebra are given here with complete proofs, an d som e definitions fro m th e abov e list o f prerequisites ar e recollected. I placed the prime emphasis on nonstandard neat proofs of known theorems. In this book I only consider finite dimensional linear spaces. The exposition is mostly performed over the fields of real or complex numbers. The peculiarity of the fields of finite characteristics is mentioned when needed. Cross-references insid e th e boo k ar e natural : 36. 2 means subsectio n 2 o f §36 ; Problem 36.2 is Problem 2 from §36 ; Theorem 36.2.2 stands for Theorem 2 from 36.2. ACKNOWLEDGMENTS. Thi s book i s based o n a course I rea d a t the Independen t University of Moscow, 1991/92. I am thankful to the participants for comments and to D. V. Beklemishev, D. B. Fuchs, A. I. Kostrikin, V. S. Retakh, A. N. Rudakov, and A. P. Veselov for fruitful discussion s of the manuscript. For th e secon d printin g (1996) , th e autho r adde d th e proo f o f th e Kronecke r theorem for pairs of linear maps (Sec. 12.6) and corrected several small errors.
XV This page intentionally left blank MAIN NOTATION S AN D CONVENTION S
denotes a matrix of size m x n\ we say that a square n x n am\ .. . matrix is of order n;
atjf sometime s denote d b y a/ y fo r clarity , i s the elemen t o r th e entr y fro m th e intersection of the /th row and the yth column; (fl/y) is another notation for the matrix A; \\a(j || stil l another notation for the matrix (0,7), where p
diagUi,..., kn) i s the diagonal matrix o f size n x n with elements an = A, - and zero offdiagonal elements ; / = diag(l,... , 1 ) is the unit matrix; whe n its size , n x « , i s needed explicitl y we denote the matrix by /„; the matrix al', where a i s a number, is called a scalar matrix ; AT i s the transposed o f A, A T = (0,7) , where a[j = ay/ ;
A = (a';;), where ajy = ~a[j\ —T A* =A ; 0- = (fc 1*"^ ) is a permutation: a(i) = /:/ ; the permutation ( ^ " ^ ) is often abbrevi - ated to (k\ >..k n); / ,x^ r f 1 i f < r is even, sign a = (-l) ff = ^ ' I -1 i f c r is odd;
Span(ei,..., en) i s the linear space spanned by the vectors e\ 9...,en. n W Given bases e\,...9en an d £1,..., em i n spaces V an d JF , respectively, we assign XV1U MAIN NOTATIONS AND CONVENTION S
*1 to a matrix A the operator A : V n — • W m whic h sends the vector ( ] into the
xn, y\ \ { &\\ •• • a\ n\ f x\ vector
Kym/ \#m l •• • o, mnJ \x n, n Since yt = ]T ) aijxb w e ^ ave
ijXjSi'y , /=! / 1= 1 /= 1 in particular, Aej = ^ at jet; i everywhere except for §3 7 the notations A > 0 , A > 0 , A < 0 , or A < 0 mean that a real symmetric or Hermitian matrix A is positive definite, nonnegative definite, negative definite, or nonpositive definite, respectively; A > B means that A — B > 0; whereas in §37 they mean that a^ > Ofor all i, j, etc. Card M i s the cardinality of the set M, i.e , the number of elements of M ; A\w denote s th e restrictio n o f th e operato r A : V — • V ont o th e subspac e W CV; sup is the least upper bound (supremum); Z, Q, R, C, H, O denote, a s usual, th e set s of al l integer, rational , real , complex , quaternion and octonion numbers, respectively; denotes the set of all positive integers (without 0);
t 0 otherwise . CHAPTER I
DETERMINANTS
The notion o f a determinant appeare d a t th e en d o f the 17t h century i n work s of Leibni z (1646-1716 ) an d a Japanes e mathematician , Sek i Kova , als o know n a s Takakazu (1642-1708). Leibni z did not publish the results of his studies related to determinants. The best known is his letter to l'Hospital (1693) in which Leibniz writes down the determinant condition of compatibility for a system of three linear equations in two unknowns. Leibniz particularly emphasized the usefulness of two indices when expressing the coefficients o f the equations. I n modern terms he actually wrote about the indices /, j i n the expression x,- = ]) P • a ijyr Seki arrived at the notion o f a determinant whil e solving the problem o f finding common roots of algebraic equations. In Europe, the search for common roots of algebraic equations soon also became the main trend associated with determinants. Newton, Bezout, and Euler studied this problem. Seki did not have the general notion of the derivative at his disposal, but he actually got an algebraic expression equivalent to the derivative of a polynomial. H e searched for multipl e roots o f a polynomial f(x) a s common roots o f f(x) an d f'{x). T o find commo n roots of polynomials f(x) an d g{x) (fo r / an d g of small degrees) Seki got determinant expressions. Th e main treatise by Seki was published in 1674 ; there applications of the method are published, rather than the method itself. H e kept the main method secret confiding only in his closest pupils. In Europe, the first publication related to determinants, due to Cramer, appeared in 1750. In this work Cramer gave a determinant expression for a solution of the problem of finding the conic through 5 fixed points (this problem reduces to a system of linear equations). The general theorems on determinants wer e proved onl y ad hoc when they were needed to solve some other problem. Therefore, the theory of determinants developed slowly, lef t behin d a s compared wit h th e genera l developmen t o f mathematics. A systematic presentation o f the theory o f determinants i s mainly associated with the names of Cauchy (1789-1857) and Jacobi (1804-1851). 1. Basi c properties of determinants The determinant of a square matrix A = ||a/ y ||" is the alternated sum
22(~^)aa\a(\)a2a(2) • • • a n
1 220 APPENDIX over C the polynomials / an d g have a nontrivial common divisor and, therefore, / and g have a nontrivial common divisor, r, ove r Q as well. Sinc e / i s an irreducible polynomial with the leading coefficient 1 , it follows that r = ±f, • A.2. THEORE M (Eisenstein's criterion). Let
n f(x) = a 0 + a\x H \- a nx be a polynomial with integer coefficients and let p be a prime such that the coefficient an 2 is not divisible by p whereas ao, ..., a n_\ are, and ao is not divisible by p . Then the polynomial f is irreducible over Z. xk c l PROOF. Suppos e that / = gh = (52bk )(52 ix )> wher e g and h are not con- stants. Th e number boco = ao is divisible by p and, therefore, one of the numbers bo or co is divisible by p. Let , for definiteness sake , b0 be divisible by p. The n c0 is not divisible by p because ao = boco is not divisible by p2. I f all numbers bf are divisible by p, the n a n i s divisible by p. Therefore , b( is not divisibl e by p fo r som e / with 0 < i < degg < n. We may assume that i is the least index for which the number 6/ is nondivisible by p. O n the one hand, by the assumption, the number at i s divisible by p. O n the other hand,
A.3. THEOREM . Suppose the numbers
(1) (ai 1} (1) (a 1} v, v v ~ v v v "" y\> y\ J • • • > y\ > •••? yn> yn > • • • > yn are given at points x\,... ,x n andm = ct\ H h aw — 1. Then there exists a polynomial Hm(x) of degree not greater than mfor which Hm(xj) = yj andH^ixj) = yy.
PROOF. Le t k = max(ai,... , an). Fo r k = 1 w e can make use of the Lagrange interpolation polynomial
L (x) = Y^ (x -xi)...{x- Xj_i){x - x y+1) ...{x-Xn)
~ {Xj - Xi ) . . . (*y - X 7-_!)(xy - Xy +1) . . . (xy - X n) ' *
Let con (x) = ( x - xj).. . ( x - x n). Tak e an arbitrary polynomial Hm-n o f degree not greater than m-n an d assign to it the polynomial Hm (x) = L n {x)+ca n (x)Hm-n (x) . It is clear that Hm (xy ) = yj fo r any polynomial Hm^.n. Besides ,
H'm(x) = L' n(x) + co'n(x)Hm-n{x) + cD n(x)H'm_n{x), APPENDIX 221 i.e.,i/^(jty) = Zjj(x/)4-o>^(x 7)iyw_rt(xy). Sincec^(xy ) ^ 0 , then at points where the values of H'm (x y ) are given, we may determine the corresponding values of Hm-n (XJ ). Furthermore,
H^(xj) = L%(xj) + a>Z(x,.)H m-n(x,) + 2co' n(Xj)H^n(xj).
Therefore, a t point s wher e th e value s o f H^(XJ) ar e give n w e ca n determin e th e corresponding values of #^_w(xy), etc. Thus, our problem reduces to the construction of a polynomial Hm-n{x) o f degree not greater than m-n fo r which H^_n{xj) = zV for / = 0,..., OLJ — 2 (i f otj = 1 , then there are no restrictions on the values of H m_n and its derivatives at xy). It is also clear that m - n = Yl( ai ~ 1 ) ~ 1 - After fc - 1 of similar operations it remains to construct Lagrange's interpolation polynomial. • A.4. Hilbert' s Nullstellensatz . W e wil l onl y nee d th e followin g specia l cas e o f Hilbert's Nullstellensatz.
THEOREM. Let f\,... , fr be polynomials in n indeterminates over C without common zeros. Then there exist polynomials g\, .. . , gr such that f\g\ H h frgr = 1 .
PROOF. Le t J(/i,..., fr) b e the ideal of the polynomial ring C [x\,..., x n] — K generated by /i,... , fr. Suppos e that there are no polynomials g\,... , gr suc h that f\g\ H V frgr — 1. Then /(/i,..., /r) 7 ^ ^- Le t / b e a nontrivial maximal ideal containing I(f\,..., /V) . A s is easy to verify, K/I i s a field. Indeed , i f / 0 7 , then I + AT / is the ideal strictly containing 7 and, therefore, this ideal coincides with K. I t follows that there exist polynomials g e K an d h e I suc h that 1 —h-^fg. The n the class ~g e K/I i s the inverse offe K/I. Now, let us prove that the field A = K/I coincide s with C. Let a/ b e the image of X( unde r the natural projectio n
p :C[xi,...,x n] — >C[xi,...,xn]/I = A.
Then A = {Yl zti---i*ai'--ain I z'i-'» G C} =C[ai,...,a„].
a a a Further, let ^0 = C and A s = C[ai,.. . ,a s]. The n A s+\ = i52 i s+\\ i € A s} = As[as+\]. Le t u s prov e b y inductio n o n s tha t ther e exist s a rin g homomorphis m / : As — • C (whic h sends 1 to 1) . Fo r s = 0 the statement i s obvious. Now , let us show how to construct a homomorphism g : As+\ — • C from th e homomorphis m / : As — • C . Fo r this let us consider two cases. a) Th e element x = a s+\ i s transcendental ove r A s. The n fo r an y (eCwe ca n n n define a homomorphism g suc h that g(a nx -\ h #o) = f(an)£ H h /(«o). Setting ^ = 0 we get a homomorphism g suc h that g(l) = 1 . m m x b) The element J C = a.y+ 1 is algebraic over ^4.y, i.e. , bmx +b m-\x ~ -\ h^ o = 0 m for certain b t e A s. The n for alU € C such that f{b m)^ + - • • + /(&o) = 0 there is determined a homomorphism g(Ylakxk) = J2f( ak)€k whic h sends 1 to 1 . As a result we get a homomorphism h : ^4 —• C such that h (1) = 1 . It is also clear that h~ x (0) is an ideal and there are no nontrivial ideals in the field A. Hence , h is a monomorphism. Sinc e Ao = C c A an d the restriction of h to Ao is the identity map h is an isomorphism. 222 APPENDIX
Thus, w e may assume tha t a, - G C. Th e projection p map s th e polynomial fi(xu...,xn) G Ktofi(au...,an) G C. Since/i,.. .,/r G /, we have />(/*) = 0 G C. Therefore, ft{a\,..., a n) = 0. Contradiction. D
A.5. THEOREM . Le f the polynomials fi{x\,...,xn) = x™' + P/(xi,..., x„), w/zer e i = \,...,n, be such that deg P/ < ra/ a«
a X Q{x\9...,Xn) = Q*(x U...9Xn,fu...Jn)=Y* i* \^ where j\ < m\,... , jn
(s\mi + y*i) + h (snmn + jn)
1 ] 1 is maximal, this monomial can only come from the monomial x' ... x n"f* ... f„". Therefore, the coefficients o f these two monomials are equal and deg Q — m. Clearly, g(xi,.. .,x„) G /(/i,...,/w) i f and onl y if e*(xi,...,x w,/i,...,/w) is the sum of monomials for which s\ -\ + sn > 1. Besides , if P{x\,..., x n) = 1 E */i.../„-*! •••*«', where /^ < mk, then
P*(xi,..., x n,fu ...,f n) = P(x\,. .., x n).
Hence, Pg/(/i,...,/n). b) I f /i, ... , /« hav e no common zero , the n b y Hilbert's Nullstellensat z the ideal /(/I,...,/*) coincide s wit h the entire polynomia l rin g and, therefore, P G I(f\>- • • >fn)', this contradicts part a). Therefore, the given system of equations is solv- able. Le t £ = (£i,..., £n) be a solution of this system. The n £™ ' = —P,-(^i, . ..,£«), where deg P, < ra;, and, therefore, any polynomial Q{£\,... £ n) can be represented in fl 1 where l the form Q(£ u...,fw) = ]£ /i ...*,» fj • • • £«" > k < ™k an d the coefficient a ivjn is the same for all solutions. Le t m = m\...m n. Th e polynomials 1 , £/, ... , £ ™ can be linearly expressed in terms of the basic monomials £{ l ... <^", where J^ < w^. Therefore, they are linearly dependent, i.e., &o+&i£/H Vbm^™ — 0, not all numbers £o> • • ->bm are zero and these numbers are the same for all solutions (do not depend on m i). Th e equation bo + b\x H h bmx = 0 has, clearly, finitely many solutions. • BIBLIOGRAPHY
Recommended literatur e R. Bellman, Introduction to matrix analysis, McGraw-Hill, New York, 1960. M. J. Crowe, A history of vector analysis, Notre Dame, London, 1967. F. R. Gantmakher, The theory of matrices, I, II, Chelsea, New York, 1959. I. M. Gelfand, Lectures on linear algebra, Interscience Tracts in Pure and Applied Math., New York, 1961. W. H. Greub, Linear algebra, Springer-Verlag, Berlin, 1967. , Multilinear algebra, Springer-Verlag, Berlin, 1967. P. R. Halmos, finite-dimensional vector spaces, Van Nostrand, Princeton, NJ, 1958. R. A. Horn and Ch. R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1986. A. I. Kostrikin and Yu. I. Manin, Linear algebra and geometry, Gordo n & Breach, New York, 1989. M. Marcus and H. Mine, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, 1964. T. Muir and W. H. Metzler, A treatise on the history of determinants, Dover, New York, 1960. M. M. Postnikov, Lectures on geometry. 2nd semester. Linear algebra., "Nauka", Moscow, 1986. (Russian) , Lectures on geometry. 5th semester. Lie groups and Lie algebras., "Mir", Moscow, 1986. G. E. Shilov, Theory of linear spaces, Prentice-Hall, Englewood Cliffs, NJ, 1961.
References J. E Adams, Vector fields on spheres, Ann. Math. 75 (1962), 603-632. S. N. Afriat, On the latent vectors and characteristic values of products of pairs of symmetric idempotents, Quart. J. Math. 7 (1956), 76-78. A. C. Aitken, A note on trace-differentiation andtheCl-operator, Proc . Edinburgh Math. Soc. 10 (1953), 1-4 . A. A. Albert, On the orthogonal equivalence of sets of real symmetric matrices, J. Math, and Mech. 7 (1958), 219-235. B. Aupetit, An improvement of Kaplansky's lemma on locally algebraic operators, Studia Math. 88 (1988), 275-278. S. Barnett, Matrices in control theory, Van Nostrand Reinhold, London, 1971. R. Bellman, Notes on matrix theory. IV, Amer. Math. Monthly 62 (1955), 172-173. R. Bellman and A. Hoffman, On a theorem of Ostrowski and Taussky, Arch. Math. 5 (1954), 123-127. M. Berger, Geometric, vol.4 (Formes quadratiques, quadriques et coniques), CEDIC/Nathan, Paris, 1977. O. I. Bogoyavlenskil, Solitons that flip over, "Nauka", Moscow, 1991 . (Russian) P.-F. Burgermeister, Classification des representation de la double fleche, L'Enseignement Math. 32 (1986), 199-210. N. N. Chan and Kim-Hung Li, Diagonal elements and eigenvalues of a real symmetric matrix, J. Math. Anal. and Appl. 91 (1983), 562-566. C. G. Cullen, A note on convergent matrices, Amer . Math. Monthly 72 (1965), 1006-1007. D. Z. Djokovic, On the Hadamard product of matrices. Math . Z. 86 (1964), 395. , Product of two involutions, Arch. Math. 18 (1967), 582-584. , A determinantal inequality for projectors in a unitary space, Proc . Amer . Math . Soc. 2 7 (1971) , 19-23. M. A. Drazin , J. W. Dungey, an d K . W . Gruenberg, Some theorems on commutative matrices, J . London Math. Soc. 26 (1951), 221-228.
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M. A . Drazin an d E . V . Haynsworth, Criteria for the reality of matrix eigenvalues, Math. Z . 7 8 (1962) , 449-452. W. N. Everitt, A note on positive definite matrices, Proc . Glasgow Math. Assoc. 3 (1958), 173-175. H. K . Faraha t an d W . Lederman , Matrices with prescribed characteristic polynomials, Proc . Edinburg h Math. Soc. 11 (1958), 143-146. H. Flanders, On spaces of linear transformations with bound rank, J. London Math. Soc. 37 (1962), 10-16 . H. Flanders and H. K. Wimmer, On matrix equations AX - XB = C and AX - YB = C , SIAM J. Appl. Math. 32 (1977), 707-710. P. Franck, Sur la meilleure approximation d'une matrice donnee par une matrice singuliere, C.R. Acad . Sci . Paris 253 (1961), 1297-1298. W. M. Frank, A bound on determinants, Proc . Amer. Math. Soc. 16 (1965), 360-363. G. Fregus, A note on matrices with zero trace, Amer. Math. Monthly 73 (1966), 630-631. Sh. Friedland, Matrices with prescribed off-diagonal elements, Israe l J. Math. 1 1 (1972), 184-189. P. M. Gibson , Matrix commutators over an algebraically closed field, Proc. Amer . Math . Soc . 5 2 (1975) , 30-32. C. Green, A multiple exchange property for bases, Proc. Amer. Math. Soc. 39 (1973), 45-50. M. J. Greenberg, Note on the Cayley-Hamilton theorem, Amer. Math. Monthly 91 (1984), 193-195. D. Yu. Grigoriev, The algebraic computational complexity of a family of bilinear forms, Zh . Vychisl. Mat. i Mat. Fiz. 19 (1979), no. 3, 563-580; English transl. in J. Comp. Math, and Math. Phys. 19 (1979). E. V. Haynsworth, Applications of an inequality for the Schur complement, Proc. Amer. Math. Soc. 24 (1970), 512-516. P. L. Hsu, On symmetric, orthogonal and skew-symmetric matrices, Proc . Edinburgh Math. Soc. 1 0 (1953), 37-44. G. H. Jacob, Another proof of the rational decomposition theorem, Amer. Math. Monthly 80 (1973), 1 Hi- ll 34. J. Kahane , Grassmann algebras for proving a theorem on Pfqffians, Linear Algebr a an d Appl . 4 (1971) , 129-139. D. C. Kleinecke, On operator commutators, Proc . Amer. Math. Soc. 8 (1957), 535-536. C. Lanczos, Linear systems in self adjoint form, Amer . Math. Monthly 65 (1958), 665-679. K. N. Majindar, On simultaneous Hermitian congruence transformations of matrices, Amer . Math. Monthly 70 (1963), 842-844. S. V. Manakov, A remark on integration of the Euler equation for an N-dimensional solid body, Funktsional . Analiz i ego Prilozhen. 1 0 (1976), no. 4, 93-94; English transl. in Functional Anal. Appl. 10 (1976). M. Marcus and H. Mine, On two theorems of Frobenius, Pac. J. Math. 60 (1975), 149-151. [a] M . Marcus and B . N. Moyls, Linear transformations on algebras of matrices, Can . J . Math. 1 1 (1959), 61-66. [b] , Transformations on tensor product spaces, Pac. J. Math. 9 (1959), 1215-1222. M. Marcus and R. Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Can . J. Math. 1 1 (1959), 383-396. W, S. Massey, Cross products of vectors in higher dimensional Euclidean spaces, Amer . Math . Monthl y 9 0 (1983), 697-701. R. Merris, Equality of decomposable symmetrized tensors, Can. J. Math. 27 (1975), 1022-1024 . L. Mirsky, An inequality for positive definite matrices, Amer . Math. Monthly 62 (1955), 428-430. , On a generalization of Hadamard's determinantal inequality due to Szasz, Arch . Math . 8 (1957) , 274-275. , A trace inequality of John von Neumann, Monatsheft e fu r Math. 79 (1975), 303-306. E. Mohr, Einfaher Beweis der verallgemeinerten Determinantensatzes von Sylvester nebst einer Verscharfung, Math. Nachrichten 1 0 (1953), 257-260. E. H. Moore, General Analysis Par t I, Mem. Amer. Phil. Soc. 1 (1935), 197. R. W . Newcomb, On the simultaneous diagonalization of two semi-definite matrices, Quart . Appl. Math. 1 9 (1961), 144-146 . L. B. Nisnevich and V . I. Bryzgalov, On a problem of n-dimensional geometry, Uspekh i Mat. Nauk 8 (1953), no. 4, 169-172 . (Russian ) A. M. Ostrowski, On Schur's complement, J . Comb. Theory (A ) 1 4 (1973), 319-323. R. A. Penrose, A generalized inverse for matrices, Proc . Cambridge Phil. Soc. 51 (1955), 406-413. R. Rado, Note on generalized inverses of matrices, Proc . Cambridge Phil.Soc. 52 (1956), 600-601. BIBLIOGRAPHY 225
A. Ramakrishnan, A matrix decomposition theorem, J. Math. Anal, and Appl. 40 (1972), 36-38. M. Reid, Undergraduate algebraic geometry, Cambridg e Univ. Press, Cambridge, 1988 . Yu. B. Reshetnyak, A newproofofa theorem ofChebotarev, Uspekh i Mat. Nauk 1 0 (1955), no. 3, 155-157. (Russian) W. E. Roth, The equations AX - YB = C and AX - XB = C in matrices, Proc. Amer. Math. Soc. 3 (1952), 392-396. H. Schwert, Direct proof of Lanczos' decomposition theorem, Amer. Math. Monthly 67 (1960), 855-860. I. Sedlacek, O incidencnich maticich orientovych grafu, Casop. Pest. Mat. 84 (1959), 303-316. Z. Sidak, Opoctu kladnych prvku v mochindch nezaporne matice, Casop. Pest. Mat. 89 (1964), 28-30. M. F. Smiley, Matrix commutators, Can. J. Math. 13 (1961), 353-355. V. Strassen, Gaussian elimination is not optimal, Numerische Math. 13 (1969), 354-356. H. Valiaho , An elementary approach to the Jordan form of a matrix, Amer . Math . Monthl y 9 3 (1986) , 711-714. H. Zassenhaus, A remark on a paper ofO. Taussky, J. Math, and Mech. 10 (1961), 179-180. This page intentionally left blank SUBJECT INDEX adjoint representation, 17 6 decomposition, Schur , 86 algebra, Cayley, 180 , 183 definite, nonnegative, 99 algebra, Clifford, 18 8 derivation, 17 6 algebra exterior, 12 4 determinant, 1 algebra, Lie, 17 6 determinant, Cauchy , 3 algebra octonion, 18 3 diagonalization, simultaneous, 99 algebra of quaternions, 18 0 double, 18 0 algebra, Grassmann, 12 4 algorithm, Euclid, 219 eigenvalue, 46, 63 alternation, 12 3 eigenvector, 63 annihilator, 41 Eisenstein's criterion, 220 elementary divisors, 89 Barnett matrix, 19 4 equation, Euler, 204 basis, orthogonal, 5 1 equation, Lax, 204 basis, orthonormal, 5 1 equation, Volterra, 206 Bernoulli numbers, 24 ergodic theorem, 11 2 Bezout matrix, 19 3 Euclid's algorithm, 219 Bezoutian, 19 3 Euler equation, 204 Binet-Cauchy's formula, 1 0 expontent of a matrix, 201 bracket, 17 6 factorization, Gauss, 87 canonical form, cyclic, 80 factorization, Gram, 87 canonical form, Frobenius, 80 first integral , 204 canonical projection, 45 form bilinear, 95 Cauchy, 1 form quadratic, 95 Cauchy determinant, 3 form quadratic positive definite, 95 Cayley algebra, 18 3 form, Hermitian, 95 Cayley transformation, 10 4 form, positive definite, 95 Cayley-Hamilton's theorem, 78 form, sesquilinear, 95 characteristic polynomial, 46, 63 Fredholm alternative, 43 Chebotarev's theorem, 1 4 Frobenius block, 80 Clifford algebra , 18 8 Frobenius's inequality, 48 cofactor of a minor, 1 1 Frobenius's matrix, 4 cofactor of an element, 1 1 Frobenius-Konig's theorem, 16 3 commutator, 17 6 complex structure, 57 Gauss lemma, 219 complexification o f a linear space, 55 Gershgorin discs, 152 complexification o f an operator, 55 Gram-Schmidt orthogonalization, 5 1 conjugation, 18 0 Grassmann, 36 content of a polynomial, 219 Grassmann algebra, 12 4 convex linear combination, 48 Courant-Fischer's theorem, 97 Hadamard product, 15 7 Cramer's rule, 2 Hadamard's inequality, 14 6 cyclic block, 80 Hankel matrix, 200 Haynsworth's theorem, 1 8 decomposition, Lanczos, 86 Hermitian adjoint, 56
227 228 SUBJECT INDEX
Hermitian form, 95 matrix invertible, 1 Hermitian product, 56 matrix irreducible, 15 8 Hilbert's Nullstellensatz, 221 matrix Jordan, 69 Hoffman-Wielandt's theorem , 16 4 matrix nilpotant, 10 7 Hurwitz-Radon's theorem, 18 5 matrix nonnegative, 15 8 matrix nonsingular, 1 idempotent, 10 8 matrix normal, 10 5 image, 42 matrix orthogonal, 10 3 inequality, Hadamard, 14 6 matrix orthonormal, 5 1 inequality, Oppenheim, 15 7 matrix permutation, 72 inequality, Schur, 150 matrix positive, 15 9 inequality, Szasz, 14 6 matrix rank of, 9 inequality, Weyl, 151, 166 matrix reducible, 15 9 inertia, Sylvester's law of, 9 6 matrix skew-symmetric, 10 1 inner product, 51 matrix Sylvester, 19 2 invariant factors, 88 matrix symmetric, 95 involution, 11 2 matrix Toeplitz, 201 matrix tridiagonal, 5 Jacobi, 1 matrix Vandermonde, 3 Jacobi identity, 17 6 min-max property, 97 Jacobi's theorem, 1 3 minor, pth order, 9 Jordan basis, 69 minor, basic, 9 Jordan block, 69 minor, principal, 9 Jordan decomposition, additive, 72 Moore-Penrose's theorem, 19 6 Jordan decomposition, multiplicative, 72 multilinear map, 119 Jordan matrix, 69 Jordan's theorem, 69 nonnegative definite, 98 kernel, 42 norm Euclidean of a matrix, 15 4 Kronecker product, 12 1 norm operator of a martix, 15 3 Kronecker theorem, 73 norm spectral of a matrix, 15 3 Kronecker-Capelli's theorem, 43 normal form, Smith, 88 null space, 42 l'Hospital, 1 Lagrange interpolation polynomial, 22 0 octonion algebra, 18 3 Lagrange's theorem, 96 operator adjoint, 38 Lanczos decomposition, 86 operator contraction, 85 Laplace's theorem, 1 1 operator diagonalizable, 64 Lax differential equation , 204 operator Hermitian, 56 Lax pair, 204 operator normal, 57, 10 5 Leibniz, 1 operator semisimple, 64 Lie algebra, 17 6 operator skew-Hermitian, 5 6 Lieb's theorem, 13 1 operator unipotent, 72 operator unitary, 56 matrices commuting, 17 3 Oppenheim's inequality, 15 7 matrices similar, 68 order lexicographic, 12 7 matrices simultaneously orthogonal complement, 41 triangularizable, 17 8 orthogonal projection, 52 matrix Barnett, 19 4 matrix centrally symmetric, 68 partition of the number, 10 7 matrix circulant, 4 Pfaflfian, 13 0 matrix (classical) adjoint of , 1 1 Plucker relations, 13 4 matrix companion, 4 polar decomposition, 8 4 matrix compound, 1 3 polynomial irreducible, 219 matrix doubly stochastic, 16 2 polynomial, annihilating of a vector, 78 matrix expontent, 201 polynomial, annihilating of an matrix Frobenius, 4 operator, 77 matrix generalized inverse of, 19 6 polynomial, minimal of an operator, 77 matrix Hankel, 200 polynomial, the content of , 21 9 matrix Hermitian, 95 product, Hadamard, 15 7 SUBJECT INDEX 229 product, vector, 18 7 theorem, Haynsworth, 1 8 product, wedge, 12 4 theorem, Hoffman-Wielandt, 16 4 projection, 10 8 theorem, Hurwitz-Radon, 18 5 projection parallel to, 109 theorem, Jacobi, 1 3 theorem, Lagrange, 96 quaternion, imaginary part of, 18 1 theorem, Laplace, 1 1 quaternion, real part of, 18 1 theorem, Lieb, 131 quaternions, 18 0 theorem, Moore-Penrose, 19 6 quotient space, 45 theorem, Schur, 86, 15 7 Toda lattice, 204 range, 42 Toeplitz matrix, 201 rank of a tensor, 13 5 trace, 63 rank of an operator, 42 realification of a linear space, 55 unipotent operator, 72 realification of an operator, 55 units of a matrix ring, 89 resultant, 19 2 row (echelon) expansion, 2 Vandermonde determinant, 3 Vandermonde matrix, 3 Schur complement, 1 7 vector contravariant, 3 8 Schur's inequality, 15 0 vector covariant, 3 8 Schur's theorem, 86,157 vector extremal, 15 8 Seki Kova, 1 vector fields linearly independent, 18 7 singular values, 15 2 vector positive, 15 8 skew-symmetrization, 12 3 vector product, 18 7 Smith normal form, 88 vector product of quaternions, 18 5 snake in a matrix, 16 3 vector skew-symmetric, 68 space, dual, 38 vector symmetric, 68 space, Hermitian, 56 Volterra equation, 206 space, unitary, 56 W. R. Hamilton, 36 spectral radius, 15 3 wedge product, 12 4 Strassen's algorithm, 13 6 Weyl's inequality, 151,16 6 Sylvester matrix, 19 2 Weyl's theorem, 15 1 Sylvester's criterion, 96 Sylvester's identity, 14 , 128 Young tableau, 10 7 Sylvester's inequality, 49 Sylvester's law of inertia, 96 symmetric functions, 1 9 symmetrization, 12 3 Szasz's inequality, 14 6
Takakazu, 1 tensor decomposable, 13 2 tensor product of operators, 12 1 tensor product of vector spaces, 11 9 tensor rank, 13 5 tensor simple, 13 2 tensor skew-symmetric, 12 3 tensor split, 13 2 tensor symmetric, 12 3 tensor, convolution of , 12 1 tensor, coordinates of , 12 0 tensor, type of, 12 0 tensor, valency of, 12 0 theorem on commuting operators, 17 4 theorem, Cayley-Hamilton, 7 8 theorem, Chebotarev, 1 4 theorem, Courant-Fischer, 97 theorem, ergodic, 11 2 theorem, Frobenius-Kdnig, 16 3 Selected Titles in This Series , , , x {Continued from the front of this publication) 109 Isa o Miyadera, Nonlinear semigroups , 199 2 108 Take o Yokonuma , Tensor space s and exterio r algebra , 199 2 107 B. M. Makarov , M. G . Goluzina , A . A . Lodkin , an d A. N. Podkorytov , Selecte d problem s i n rea l analysis, 199 2 106 G.-C . Wen , Conformal mapping s an d boundar y valu e problems, 199 2 105 D . R. Yafaev , Mathematical scatterin g theory : Genera l theory , 199 2 104 R . L. Dobrushin , R . Kotecky , an d S. Shlosman , Wulf f construction : A globa l shap e fro m loca l interaction, 199 2 103 A . K . Tsikh, Multidimensional residue s and thei r applications , 199 2 102 A . M. D'in , Matching o f asymptotic expansion s o f solution s o f boundary valu e problems, 199 2 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Dong Zhen-xi , Qualitativ e theory o f differentia l equations , 199 2 100 V . L. Popov , Groups, generators , syzygies , and orbit s i n invariant theory , 199 2 99 Nori o Shimakura , Partial differentia l operator s o f elliptic type, 199 2 98 V . A. Vassiliev , Complements o f discriminants o f smoot h maps : Topolog y an d applications , 199 2 (revised edition , 1994 ) 97 Itir o Tamura , Topology o f foliations : A n introduction , 199 2 96 A , I. Markushevich , Introduction t o th e classica l theor y o f Abelian functions , 199 2 95 Guangchan g Dong , Nonlinear partia l differentia l equation s o f secon d order , 199 1 94 Yu . S . B'yashenko, Finitenes s theorem s fo r limi t cycles , 199 1 93 A . T . Fomenko an d A. A. Tuzhilin , Elements o f the geometry an d topolog y o f minimal surface s i n three-dimensional space , 199 1 92 E . M. Nikishi n an d V . N. Sorokin , Rational approximation s an d orthogonality , 199 1 91 Mamor u Mimur a an d Hiros i Toda , Topology o f Li e groups, I and II , 199 1 90 S . L. Sobolev, Some applications o f functional analysi s in mathematical physics , third edition , 199 1 89 Valer n V . Kozlov an d Dmitri! V. Treshchev, Billiards : A geneti c introductio n t o th e dynamic s o f systems with impacts, 199 1 88 A . G . Khovanskn , Fewnomials, 199 1 87 Aleksand r Robertovic h Kemer , Ideals o f identitie s o f associativ e algebras , 199 1 86 V . M. Kadet s an d M. I . Kadets, Rearrangements o f serie s in Banac h spaces , 199 1 85 Mikio Is e an d Masaru Takeuchi , Li e groups I , II, 199 1 84 Da o Tron g Th i an d A. T . Fomenko, Minimal surfaces , stratifie d multivarifolds , an d th e Platea u problem, 199 1 83 N . I. Portenko , Generalized diffusio n processes , 199 0 82 Yasutak a Sibuya , Linear differentia l equation s i n th e comple x domain : Problem s o f analyti c continuation, 199 0 81 I . M. Gelfan d an d S. G . Gindikin , Editors , Mathematica l problem s o f tomography, 199 0 80 Junjir o Noguchi an d Takushiro Ochiai , Geometri c functio n theor y i n severa l comple x variables , 1990 79 N . I. Akhiezer , Element s o f th e theor y o f ellipti c functions , 199 0 78 A . V . Skorokhod, Asymptotic method s o f th e theor y o f stochasti c differentia l equations , 198 9 77 V . M. Filippov , Variational principle s fo r nonpotentia l operators , 198 9 76 Philli p A. Griffiths , Introductio n t o algebrai c curves, 198 9 75 B . S. Kashi n an d A. A. Saakyan , Orthogonal series , 198 9 74 V . I. Yudovich , The linearizatio n metho d i n hydrodynamica l stabilit y theory , 198 9 73 Yu . G . Reshetnyak , Spac e mappings wit h bounded distortion , 198 9 72 A . V . Pogorelev, Bending s o f surface s an d stabilit y o f shells , 198 8 71 A . S. Markus, Introduction t o th e spectra l theor y o f polynomial operato r pencils , 198 8 70 N . I . Akhiezer , Lectures o n integra l transforms , 198 8 (See the AMS catalog fo r earlie r titles )