Problems and theorems in linear algebra:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
1994
|
| Schriftenreihe: | Translations of mathematical monographs
134 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006447730&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Aus dem Russ. übers. |
| Umfang: | XVIII, 225 S. |
| ISBN: | 0821802364 |
Internformat
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| 240 | 1 | 0 | |a Zadači i teoremy linejnoj algebry |
| 245 | 1 | 0 | |a Problems and theorems in linear algebra |c V. V. Prasolov |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 1994 | |
| 300 | |a XVIII, 225 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Translations of mathematical monographs |v 134 | |
| 500 | |a Aus dem Russ. übers. | ||
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| 830 | 0 | |a Translations of mathematical monographs |v 134 |w (DE-604)BV000002394 |9 134 | |
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Datensatz im Suchindex
| _version_ | 1819375521417396224 |
|---|---|
| adam_text | CONTENTS
Preface xv
Main notations and conventions xvii
Chapter I. Determinants 1
Historical remarks: Leibniz and Seki Kova. Cramer, L Hospital,
Cauchy and Jacobi
1. Basic properties of determinants 1
The Vandermonde determinant and its application. The Cauchy determinant.
Continued fractions and the determinant of a tridiagonal matrix. Certain 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 generalized Sylvester s identity. Chebotarev s theorem on the
matrix ||e |^~1, where e = exp(2ni/p).
Problems
3. The Schur complement 16
Given A = ( A[2 ], the matrix (A AU) = A22 A2lA~lA[2 is called the
A2 A22)
Schur complement (of A11 in A).
3.1. det A = det Au det (A An)
3.2. Theorem. (A B) = ((A C) {B C)).
Problems
4. Symmetric functions, sums x H hi*, and Bernoulli numbers 19
Determinant relations between ok [x ,..., xn), sk (x ,..., xn) = x + ¦ ¦ ¦ + x%
and/ty(jq,.. .,xn) = Yl x •••¦*£ • A determinant formula for Sn(k) = 1 +
i+¦¦ *=
h (k 1) . The Bernoulli numbers and Sn{k).
4.4. Theorem. Let u = S {x) and v = S2(x). Then for k 1 there exist
polynomialspi andqk such that S2A:+iU) = u2pk(u) andS2i,(x) = vqk(u).
Problems
Solutions
Chapter II. Linear spaces 35
Historical remarks: Hamilton and Grassmann
5. The dual space. The orthogonal complement 37
Linear equations and their application to the following theorem:
vii
viii CONTENTS
5.4.3. Theorem. If a rectangle with sides a and b is arbitrarily cut into squares with
sides x ,...,xn then — € Q and — € Q for all i.
a b
Problems
6. The kernel (null space) and the image (range) of an operator. The
quotient space 42
6.2.1. Theorem. KerA* = (imA)1 and mA* = (KerA) 1.
Fredholm s alternative. Kronecker Capelli s theorem. Criteria for solvability of
the matrix equation C = AXB.
Problem
7. Bases of a vector space. Linear independence 45
Change of basis. The characteristic polynomial.
7.2. Theorem. Let x ,...,xn and y [,..., yn be two bases, 1 k n. Then k of
the vectors i..... yn can be interchanged with some k of the vectors x ,..., xn so that
we get again two bases.
7.3. Theorem. Let T : V — V be a linear operator such that the vectors
Q,Tc,... ,T ( are linearly dependent for every £ 6 V. Then the operators 1, 7 ,..., T
are linearly dependent.
Problems
8. The rank of a matrix 48
The Frobenius inequality. The Sylvester inequality.
8.3. Theorem. Let U be a linear subspace of the space Mn.m ofn x m matrices, and
r m n. // rank X r for any X e U then dim U rn.
A description of subspaces U C Mn,m such that dim U = nr.
Problems
9. Subspaces. The Gram Schmidt orthogonalization process 51
Orthogonal projections.
9.5. Theorem. Let e ,...,£¦„ be an orthogonal basis for a space V, d( = e, . The
projections of the vectors e ,... ,en onto an m dimensional subspace of V have equal
lengths if and only if df(d~~ + ¦ ¦ ¦ + (/„ ) m for every i = !,...,«.
9.6.1. Theorem. A set of k dimensional subspaces of V is such that any two of these
subspaces have a common (k — ) dimensional subspace. Then either all these subspaces
have a common (k — 1 ) dimensional subspace or all of them are contained in the same
(k + ) dimensional subspace.
Problems
10. Complexification and realification. Unitary spaces 55
Unitary operators. Normal operators.
10.3.4. Theorem. Let B and C be Hermitian operators. Then the operator A =
B + iC is normal if and only if BC = CB.
Complex structures.
Problems
Solutions
Chapter III. Canonical forms of matrices and linear operators 63
11. The trace and eigenvalues of an operator 63
The eigenvalues of an Hermitian operator and of a unitary operator. The eigen¬
values of a tridiagonal matrix.
Problems
CONTENTS ix
12. The Jordan canonical (normal) form 68
12.1. Theorem. If A and B are matrices with real entries and A = PBP~ for some
matrix P with complex entries then A = QBQ1 for some matrix Q with real entries.
The existence and uniqueness of the Jordan canonical form (Valiacho s simple
proof).
The real Jordan canonical form.
12.5.1. Theorem, a) For any operator A there exist a nilpotent operator An and a
semisimple operator As such that A = As + An and AsAn — AnAs.
b) The operators An and As are unique; besides, As = S(A) and An = N(A)for
some polynomials S and N.
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.
Problems
13. The minimal polynomial and the characteristic polynomial 73
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 if and only if for any vector (x ,..., ,vn) there exist a column P and
a row Q such that xk = QAkP.
Hamilton Cayley s theorem and its generalization for polynomials of matrices.
Problems
14. The Frobenius canonical form 75
Existence of the Frobenius canonical form (H. G. Jacob s simple proof)
Problems
15. How to reduce the diagonal to a convenient form 77
15.1. Theorem. If A ^ /./ then A is similar to a matrix with the diagonal elements
(O,...,0,tr/0.
15.2. Theorem. Any matrix A is similar to a matrix with equal diagonal elements.
15.3. Theorem. Any nonzero square matrix A is similar to a matrix all diagonal
elements of which are nonzero.
Problems
16. The polar decomposition 80
The polar decomposition of noninvertible and of invertible matrices. The unique¬
ness of the polar decomposition of an invertible matrix.
16.1. Theorem. If A = S U = U2S2 are polar decompositions ofan invertible
matrix A then U = Vi.
16.2.1. Theorem. For any matrix A there exist unitary matrices U, W and a diagonal
matrix D such that A = UD W.
Problems
17. Factorizations of matrices 81
17.1. Theorem. For any complex matrix A there exist a unitary matrix V 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
x CONTENTS
18. The Smith normal form. Elementary factors of matrices 83
Problems
Solutions
Chapter IV. Matrices of special form 91
19. Symmetric and Hermitian matrices 91
Sylvester s criterion. Sylvester s law of inertia. Lagrange s theorem on quadratic
forms. Courant Fisher s theorem.
19.5.1.Theorem. If A Oand(Ax,x) = 0 for any x, then A = 0.
Problems
20. Simultaneous diagonalization of a pair of Hermitian forms 95
Simultaneous diagonalization of two Hermitian matrices A and B when A 0.
An example of two Hermitian matrices which cannot be simultaneously diagonalized.
Simultaneous diagonalization of two semidefinite matrices. Simultaneous diagonal¬
ization of two Hermitian matrices A and B such that there is no x ^ 0 for which
x Ax = x Bx = 0.
Problems
21. Skew symmetric matrices 97
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
^2,(xik y2k X2k 2k l)
Jfc=l
Problems
22. Orthogonal matrices. The Cayley transformation 99
The standard Cayley transformation of an orthogonal matrix which does not have
1 as its eigenvalue. The generalized Cayley transformation of an orthogonal matrix
which has 1 as its eigenvalue.
Problems
23. Normal matrices 101
23.1.1. Theorem. If an operator A is normal then Ker A* = Ker A and Im A* =
mA.
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 103
24.2.1. Theorem. Let A be an n x n matrix. The matrix A is nilpotent if and only if
tr(/C) = Ofor each p = 1,...,«.
Nilpotent matrices and Young tableaux.
Problems
25. Projections. Idempotent matrices 104
25.2.I 2. Theorem. An idempotent operator P is an Hermitian one if and only if a)
Ker P _L Im P; or b) Px x for every x.
CONTENTS xi
25.2.3. Theorem. Let P ,... ,Pn be Hermitian, idempotent operators. The operator
P = P + ¦ ¦ ¦ + Pnisan idempotent one if and only ifPiPj = 0 whenever i / j.
25.4.1. Theorem. Let V ffi ¦ ¦ ¦ ffi Vk,P, : V — K, be Hermitian idempotent
operators, A = P + ¦ ¦ ¦ + Pk. Then 0 det A 1 and det A = 1 if and only if
Vj A. Vj whenever i £ j.
Problems
26. Involutions 108
26.2. Theorem. A matrix A can be represented as the product of two involutions if
and only if the matrices A and A ~ are similar.
Problems
Solutions
Chapter V. Multilinear algebra 115
27. Multilinear maps and tensor products 115
An invariant definition of the trace. Kronecker s product of matrices. A 0 B: the
eigenvalues of the matrices A ® B and A®I + I®B. Matrix equations AX XB =
C andAX XB = IX.
Problems
28. Symmetric and skew symmetric tensors 119
The Grassmann algebra. Certain canonical isomorphisms. Applications of Grass
mann algebra: proofs of Binet Cauchy s formula and Sylvester s identity.
28.5.4. Theorem. Let AB(t) = 1 + £ tr(A«)r« and SB(t) = 1 + E tr(S«)r«.
Then SB(t) = (AB( t))~K
Problems
29. ThePfaffian 125
The Pfaffian of principal submatrices of the matrix M = || ii7||| ¦ where m,; =
29.2.2. Theorem. Given a skew symmetric matrix A we have
Pf(A + SM) = ±^pk,Wherepk=j:A(ll ; *«• * ).
Problems
30. Decomposable skew symmetric and symmetric tensors 128
30.1.1. Theorem. x A ¦ Axk = y A ¦ Ayk ^ 0 if and only if Span(x ,. ..,xk) =
Span( ¦),...,»).
30.1.2. Theorem. S(x ® ¦ ¦ ¦ g xk) = S(vi 8 • ¦ ¦ ® yk) ^ 0 if and only if
SpanUi,...,^) = Span( i,...,vt).
Plucker relations.
Problems
31. The tensor rank 131
Strassen s algorithm. The set of all tensors of rank 2 is not closed. The rank
over K is not equal, generally, to the rank over C.
Problems
xii CONTENTS
32. Linear transformations of tensor products 133
A complete description of the following types of transformations of
Vm®(V) ^ Mm,n:
1) rank preserving:
2) determinant preserving;
3) eigenvalue preserving;
4) invertibility preserving.
Problems
Solutions
Chapter VI. Matrix inequalities 141
33. Inequalities for symmetric and Hermitian matrices 141
33.1.1. Theorem. IfA B 0 then A 1 B~K
33.1.3. Theorem. If A 0 is a real matrix then
(A~xx,x) = max(2(x,y) (Ay,y)).
33.2.1. Theorem. Suppose A = ( i[ f ) 0. Then A At ¦ A2 .
B Ai}
Hadamard s inequality and Szasz s inequality.
n
33.3.1. Theorem. Suppose a, 0, J] a, = 1 and Aj 0. Then
a A + + akAk Axp + ¦ ¦ ¦ + Ak a .
33.3.2. Theorem. Suppose A, 0, a, e C Then
|det(ai^i H V akAk) dex{ a A H h ak Ak).
Problems
34. Inequalities for eigenvalues 146
Schur s inequality. Weyl s inequality (for eigenvalues of A + B).
(B C
] 0 be an Hermitian matrix, ct ¦ ¦ ¦
C B)
an and fi ¦ ¦ ¦ fim the eigenvalues of A and B, respectively. Then a, Pi
an i —m
34.3. Theorem. Let A and B be Hermitian idempotents, /. any eigenvalue of AB.
ThenO /. 1.
34.4.1. Theorem. Let the /., andni be the eigenvalues of A and AA*. respectively; let
ni = y/Wi Let |/.[ ¦ ¦ • /.„, where n is the order of A. Then /. ... km o ... am.
34.4.2.Theorem. Let o • ¦ ¦ an and T ¦ ¦ ¦ tn be the singular values of A
andB. Then u(AB) Y.°i?i
Problems
35. Inequalities for matrix norms 149
The spectral norm A S and the Euclidean norm ||^||,,. the spectral radius p(A).
35.1.2. Theorem. If a matrix A is normal then p(A) = A S.
35.2. Theorem. A S A e Jh A S.
The invariance of the matrix norm and singular values.
CONTENTS xiii
A + A*
35.3.1. Theorem. Let S be an Hermitian matrix. Then A |j does not
exceed A — 5||, 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 A — U e A — W eandif A ^ 0, then the equality is only attained
for W = U.
Problems
36. Schur s complement and Hadamard s product. Theorems of
Emily Haynsworth 151
36.1.1. Theorem. If A 0 then (A A {) 0.
36.1.4. Theorem. If A^ and B^ are the kthprincipalsubmatrices of positive definite
order n matrices A and B, then
M+SISMI(,+gm)+l l(l+!^).
Hadamard s product A o B.
36.2.1. Theorem. If A Oand B 0 then A o B 0.
Oppenheim s inequality
Problems
37. Nonnegative matrices 154
Wielandt s theorem
Problems
38. Doubly stochastic matrices 158
Birkhoffs theorem. H.Weyl s inequality.
Solutions
Chapter VII. Matrices in algebra and calculus 169
39. Commuting matrices 169
The space of solutions of the equation AX = XA for X with the given A of order
n.
39.2.2. Theorem. Any set of commuting diagonalizable operators has a common
eigenbasis.
39.3. Theorem. Let A,B be matrices such that AX = XA implies BX = XB.
Then B = g(A), where g is a polynomial.
Problems
40. Commutators 171
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 ndsx B = Ofor
somes 0. Then B = g (A) for a polynomial g.
40.4. Theorem. Matrices A ,...,An can be simultaneously triangularized over C
if and only if the matrix p(A ,... ,An)[Aj,Aj] is a nilpotent one for any polynomial
p(x ,..., xn) in noncommuting indeterminates.
40.5. Theorem. Ifrank[A,B] 1, then A and B can be simultaneously triangular¬
ized over C.
Problems
xiv CONTENTS
41. Quaternions and Cayley numbers. Clifford algebras 176
Isomorphisms so(3,K) = su(2) and so(4, M) = so(3,K) ® so(3,lR). The vector
products in R3 andR7. Hurwitz Radon families of matrices. Hurwitz Radon number
p(2c+4d(2a + 1)) = 2C + d.
41.7.1. Theorem. The identity of the form
(Xj + ...+X2)(yj + ...+y2) = {:2 + ...+:2l
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 xn matrices, a subspace of invertible matrices
of dimension m exists if and only ifm p{n).
Other applications: algebras with norm, vector product, linear vector fields on
spheres.
Clifford algebras and Clifford modules.
Problems
42. Representations of matrix algebras 186
Complete reducibility of finite dimensional representations of Mat( V).
Problems
43. The resultant 187
Sylvester s matrix. Bezout s matrix and Barnett s matrix
Problems
44. The general inverse matrix. Matrix equations 192
44.3. Theorem, a) The equation AX — XA = C is solvable if and only if the
. (A O , (A C ...
matrices I I and I 1 are similar.
b) The equation AX — YA = C is solvable if and only if rank I I =
O B j
Ho «)
Problems
45. Hankel matrices and rational functions 195
46. Functions of matrices. Differentiation of matrices 197
Differential equation X = AX and the Jacobi formula for det A.
Problems
47. Lax pairs and integrable systems 200
48. Matrices with prescribed eigenvalues 202
48.1.2. Theorem. For any polynomial f(x) = x + c xn~l + ¦ ¦ ¦ + 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 ,...,xn so that the eigenvalues of A are given complex
numbers; there are finitely many sets {x ,...,xn} satisfying this condition.
Solutions
Appendix 215
Eisenstein s criterion, Hilbert s Nullstellensatz.
Bibliography 219
Subject Index 223
|
| any_adam_object | 1 |
| author | Prasolov, Viktor V. 1956- |
| author_GND | (DE-588)129271977 |
| author_facet | Prasolov, Viktor V. 1956- |
| author_role | aut |
| author_sort | Prasolov, Viktor V. 1956- |
| author_variant | v v p vv vvp |
| building | Verbundindex |
| bvnumber | BV009747933 |
| classification_rvk | SK 220 |
| ctrlnum | (OCoLC)246829017 (DE-599)BVBBV009747933 |
| discipline | Mathematik |
| format | Book |
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| genre | (DE-588)4143389-0 Aufgabensammlung gnd-content |
| genre_facet | Aufgabensammlung |
| id | DE-604.BV009747933 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T09:41:33Z |
| institution | BVB |
| isbn | 0821802364 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006447730 |
| oclc_num | 246829017 |
| open_access_boolean | |
| owner | DE-12 DE-20 DE-824 |
| owner_facet | DE-12 DE-20 DE-824 |
| physical | XVIII, 225 S. |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spellingShingle | Prasolov, Viktor V. 1956- Problems and theorems in linear algebra Translations of mathematical monographs Theorem (DE-588)4185093-2 gnd Lineare Algebra (DE-588)4035811-2 gnd Matrizenrechnung (DE-588)4126963-9 gnd |
| subject_GND | (DE-588)4185093-2 (DE-588)4035811-2 (DE-588)4126963-9 (DE-588)4143389-0 |
| title | Problems and theorems in linear algebra |
| title_alt | Zadači i teoremy linejnoj algebry |
| title_auth | Problems and theorems in linear algebra |
| title_exact_search | Problems and theorems in linear algebra |
| title_full | Problems and theorems in linear algebra V. V. Prasolov |
| title_fullStr | Problems and theorems in linear algebra V. V. Prasolov |
| title_full_unstemmed | Problems and theorems in linear algebra V. V. Prasolov |
| title_short | Problems and theorems in linear algebra |
| title_sort | problems and theorems in linear algebra |
| topic | Theorem (DE-588)4185093-2 gnd Lineare Algebra (DE-588)4035811-2 gnd Matrizenrechnung (DE-588)4126963-9 gnd |
| topic_facet | Theorem Lineare Algebra Matrizenrechnung Aufgabensammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006447730&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
| work_keys_str_mv | AT prasolovviktorv zadaciiteoremylinejnojalgebry AT prasolovviktorv problemsandtheoremsinlinearalgebra |