Problems and theorems in linear algebra

228 Pages·1994·1.721 MB·English

by  V. V. Prasolov

PROBLEMS AND THEOREMS
IN LINEAR ALGEBRA
V. Prasolov
Abstract. Thisbookcontainsthebasicsoflinearalgebrawithanemphasisonnon-
standardandneatproofsofknowntheorems. Manyofthetheoremsoflinearalgebra
obtained mainly during the past 30 years are usually ignored in text-books but are
quiteaccessibleforstudentsmajoringorminoringinmathematics. Thesetheorems
aregivenwithcompleteproofs. Thereareabout230problemswithsolutions.
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1
CONTENTS
Preface
Main notations and conventions
Chapter I. Determinants
Historical remarks: Leibniz and Seki Kova. Cramer, L’Hospital,
Cauchy and Jacobi
1. Basic properties of determinants
The Vandermonde determinant and its application. The Cauchy deter-
minant. Continued fractions and the determinant of a tridiagonal matrix.
Certainotherdeterminants.
Problems
2. Minors and cofactors
Binet-Cauchy’sformula. Laplace’s theorem. Jacobi’s theorem on minors
of the adjoint matrix. The generalized Sylvester’s identity. Chebotarev’s
ř ř
theoremonthematrixřεijřp−1,whereε=exp(2πi/p).
1
Problems
3. The Schur complement
ţ ű
Given A = AA1211 AA1222 , the matrix (A|A11) = A22−A21A−111A12 is
calledthe Schur complement(ofA11 inA).
3.1. det A=det A11det (A|A11).
3.2. Theorem. (A|B)=((A|C)|(B|C)).
Problems
4. Symmetricfunctions,sumsxk+···+xk,andBernoullinumbers
1 n
DeterminantrelationsbetwPeenσk(x1,...,xn),sk(x1,...,xn)=xk1+···+
xkn and pk(x1,...,xn) = xi11...xinn. A determinant formula for
i1+...ik=n
Sn(k)=1n+···+(k−1)n. TheBernoullinumbersandSn(k).
4.4. Theorem. Letu=S1(x)andv=S2(x). Thenfork≥1thereexist
polynomials pk and qk such that S2k+1(x)=u2pk(u) and S2k(x)=vqk(u).
Problems
Solutions
Chapter II. Linear spaces
Historical remarks: Hamilton and Grassmann
5. The dual space. The orthogonal complement
Linearequationsandtheirapplicationtothefollowingtheorem:
5.4.3. Theorem. Ifarectanglewithsidesaandbisarbitrarilycutinto
squares with sides x1,...,xn then xi ∈Q and xi ∈Q for all i.
a b
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2
Problems
6. The kernel (null space) and the image (range) of an operator.
The quotient space
6.2.1. Theorem. KerA∗=(ImA)⊥ and ImA∗=(KerA)⊥.
Fredholm’s alternative. Kronecker-Capelli’s theorem. Criteria for solv-
abilityofthematrixequationC=AXB.
Problem
7. Bases of a vector space. Linear independence
Changeofbasis. Thecharacteristicpolynomial.
7.2. Theorem. Let x1,...,xn and y1,...,yn be two bases, 1≤k≤n.
Then k of the vectors y1,...,yn can be interchanged with some k of the
vectors x1,...,xn so that we get again two bases.
7.3. Theorem. Let T : V −→ V be a linear operator such that the
vectors ξ,Tξ,...,Tnξ are linearly dependent for every ξ ∈ V. Then the
operators I,T,...,Tn are linearly dependent.
Problems
8. The rank of a matrix
TheFrobeniusinequality. TheSylvesterinequality.
8.3. Theorem. Let U be a linear subspace of the space Mn,m of n×m
matrices, and r≤m≤n. If rankX≤r for any X∈U then dimU ≤rn.
AdescriptionofsubspacesU ⊂Mn,m suchthatdimU =nr.
Problems
9. Subspaces. The Gram-Schmidt orthogonalization process
Orthogonalprojections.
9.ř5. řTheorem. Let e1,...,en be an orthogonal basis for a space V,
di=řeiř. The projections of the vectors e1,...,en onto an m-dimensional
subspace of V have equal lengths if and only if d2i(d−12+···+d−n2)≥m for
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.
Theneitherallthesesubspaceshaveacommon(k−1)-dimensionalsubspace
or all of them are contained in the same (k+1)-dimensional subspace.
Problems
10. Complexification and realification. Unitary spaces
Unitaryoperators. Normaloperators.
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.
Complexstructures.
Problems
Solutions
Chapter III. Canonical forms of matrices and linear op-
erators
11. The trace and eigenvalues of an operator
TheeigenvaluesofanHermitianoperatorandofaunitaryoperator. The
eigenvaluesofatridiagonalmatrix.
Problems
12. The Jordan canonical (normal) form
12.1. Theorem. If A and B are matrices with real entries and A =
PBP−1 forsomematrixP withcomplexentriesthenA=QBQ−1 forsome
matrix Q with real entries.
CONTENTS 3
The existence and uniqueness of the Jordan canonical form (V¨aliacho’s
simpleproof).
TherealJordancanonicalform.
12.5.1. Theorem. a)ForanyoperatorAthereexistanilpotentoperator
AnandasemisimpleoperatorAssuchthatA=As+AnandAsAn=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. ForanyinvertibleoperatorAthereexistaunipotent
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
13.1.2. Theorem. For any operator A there exists a vector v such that
the minimal polynomial of v (with respect to A) coincides with the minimal
polynomial of A.
13.3. Theorem. The characteristic polynomial of a matrixA coincides
with its minimal polynomial if and only if for any vector (x1,...,xn) there
exist a column P and a row Q such that xk=QAkP.
Hamilton-Cayley’stheoremanditsgeneralizationforpolynomialsofma-
trices.
Problems
14. The Frobenius canonical form
ExistenceofFrobenius’scanonicalform(H.G.Jacob’ssimpleproof)
Problems
15. How to reduce the diagonal to a convenient form
15.1. Theorem. If A 6= λI then A is similar to a matrix with the
diagonal elements (0,...,0,trA).
15.2. Theorem. AnymatrixAissimilartoamatrixwithequaldiagonal
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
Thepolardecompositionofnoninvertibleandofinvertiblematrices. The
uniquenessofthepolardecompositionofaninvertiblematrix.
16.1. Theorem. If A = S1U1 = U2S2 are polar decompositions of an
invertible matrix A then U1=U2.
16.2.1. Theorem. For any matrix A there exist unitary matrices U,W
and a diagonal matrix D such that A=UDW.
Problems
17. Factorizations of matrices
17.1. Theorem. ForanycomplexmatrixAthereexistaunitarymatrix
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’s,andLanczos’factorizations.
17.3. Theorem. Anymatrixisaproductoftwosymmetricmatrices.
Problems
18. Smith’s normal form. Elementary factors of matrices
Problems
Solutions
4
Chapter IV. Matrices of special form
19. Symmetric and Hermitian matrices
Sylvester’s criterion. Sylvester’s law of inertia. Lagrange’s theorem on
quadraticforms. Courant-Fisher’stheorem.
19.5.1.Theorem. If A≥0 and (Ax,x)=0 for any x, then A=0.
Problems
20. Simultaneous diagonalization of a pair of Hermitian forms
Simultaneous diagonalization of two Hermitian matrices A and B when
A>0. AnexampleoftwoHermitianmatriceswhichcannotbesimultane-
ously diagonalized. Simultaneous diagonalization of two semidefinite matri-
ces. Simultaneous diagonalization of two Hermitian matrices A and B such
thatthereisnox6=0forwhichx∗Ax=x∗Bx=0.
Problems
§21. Skew-symmetric 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)=0 for all x,
then A is a skew-symmetric matrix.
21.2. Theorem. Anyskew-symmetricbilinearformcanbeexpressedas
Pr
(x2k−1y2k−x2ky2k−1).
k=1
Problems
22. Orthogonal matrices. The Cayley transformation
ThestandardCayleytransformationofanorthogonalmatrixwhichdoes
not have 1 as its eigenvalue. The generalized Cayley transformation of an
orthogonalmatrixwhichhas1asitseigenvalue.
Problems
23. Normal matrices
23.1.1. Theorem. IfanoperatorAisnormalthenKerA∗=KerAand
ImA∗=ImA.
23.1.2. Theorem. An operator A is normal if and only if any eigen-
vector of A is an eigenvector of A∗.
23.2. Theorem. If an operator A is normal then there exists a polyno-
mial P such that A∗=P(A).
Problems
24. Nilpotent matrices
24.2.1. Theorem. LetAbeann×nmatrix. ThematrixAisnilpotent
if and only if tr(Ap)=0 for each p=1,...,n.
NilpotentmatricesandYoungtableaux.
Problems
25. Projections. Idempotent matrices
25.2.1&2. Theorem. An idempotent operator P is an Hermitian one
if and only if a) KerP ⊥ImP; or b) |Px|≤|x| for every x.
25.2.3. Theorem. Let P1,...,Pn be Hermitian, idempotent operators.
TheoperatorP =P1+···+Pn isanidempotentoneifandonlyifPiPj =0
whenever i6=j.
25.4.1. Theorem. Let V1 ⊕···⊕Vk,Pi : V −→ Vi be Hermitian
idempotentoperators,A=P1+···+Pk. Then0<detA≤1anddetA=1
if and only if Vi⊥Vj whenever i6=j.
Problems
26. Involutions
CONTENTS 5
26.2. Theorem. A matrix A can be represented as the product of two
involutions if and only if the matrices A and A−1 are similar.
Problems
Solutions
Chapter V. Multilinear algebra
27. Multilinear maps and tensor products
An invariant definition of the trace. Kronecker’s product of matrices,
A⊗B; the eigenvalues of the matrices A⊗B and A⊗I+I⊗B. Matrix
equationsAX−XB=CandAX−XB=λX.
Problems
28. Symmetric and skew-symmetric tensors
The Grassmann algebra. Certain canonical isomorphisms. Applications
ofGrassmannalgebra: proofsofBinet-Cauchy’sformulaandSylvester’siden-
tity.
Pn
28.5.4. Theorem. Let ΛB(t) = 1+ tr(ΛqB)tq and SB(t) = 1+
q=1
Pn
tr(SBq)tq. Then SB(t)=(ΛB(−t))−1.
q=1
Problems
29. The Pfaffian
ř ř
ThePfaffianofprincipalsubmatricesofthematrixM =řmijř21n,where
mij =(−1)i+j+1.
29.2.2. Theorem. Given a skew-symmetric matrix A we have
à !
Pf(A+λ2M)=kX=n0λ2kpk, wherepk=Xσ A σσ11 ...... σσ22((nn−−kk))
Problems
30. Decomposable skew-symmetric and symmetric tensors
30.1.1. Theorem. x1 ∧···∧xk = y1 ∧···∧yk 6= 0 if and only if
Span(x1,...,xk)=Span(y1,...,yk).
30.1.2. Theorem. S(x1⊗···⊗xk)=S(y1⊗···⊗yk)6=0 if and only
if Span(x1,...,xk)=Span(y1,...,yk).
Plu¨ckerrelations.
Problems
31. The tensor rank
Strassen’salgorithm. Thesetofalltensorsofrank≤2isnotclosed. The
rankoverRisnotequal,generally,totherankoverC.
Problems
32. Linear transformations of tensor products
Acompletedescriptionofthefollowingtypesoftransformationsof
Vm⊗(V∗)n∼=Mm,n:
1)rank-preserving;
2)determinant-preserving;
3)eigenvalue-preserving;
4)invertibility-preserving.
6
Problems
Solutions
Chapter VI. Matrix inequalities
33. Inequalities for symmetric and Hermitian matrices
33.1.1. Theorem. If A>B>0 then A−1<B−1.
33.1.3. Theorem. If A>0 is a real matrix then
(A−1x,x)=max(2(x,y)−(Ay,y)).
y
ţ ű
33.2.1. Theorem. Suppose A = BA1∗ AB2 > 0. Then |A| ≤ |A1|·
|A2|.
Hadamard’sinequalityandSzasz’sinequality.
Pn
33.3.1. Theorem. Suppose αi>0, αi=1 and Ai>0. Then
i=1
|α1A1+···+αkAk|≥|A1|α1+···+|Ak|αk.
33.3.2. Theorem. Suppose Ai≥0,αi∈C. Then
|det(α1A1+···+αkAk)|≤det(|α1|A1+···+|αk|Ak).
Problems
34. Inequalities for eigenvalues
Schur’sinequality. Weyl’sinequality(foreigenvaluesofA+B).
ţ ű
B C
34.2.2. Theorem. Let A = > 0 be an Hermitian matrix,
C∗ B
α1≤···≤αn and β1≤···≤βm the eigenvalues of A and B, respectively.
Then αi≤βi≤αn+i−m.
34.3. Theorem. Let A and B be Hermitian idempotents, λ any eigen-
value of AB. Then 0≤λ≤1.
34.4.1. Theorem√. Let the λi and µi be the eigenvalues of A and AA∗,
respectively; let σi = µi. Let |λ1 ≤ ··· ≤ λn, where n is the order of A.
Then |λ1...λm|≤σ1...σm.
34.4.2.Theorem. Let σ1 ≥···≥Pσn and τ1 ≥···≥τn be the singular
values of A and B. Then |tr(AB)|≤ σiτi.
Problems
35. Inequalities for matrix norms
ThespectralnormkAk andtheEuclideannormkAk ,thespectralradius
s e
ρ(A).
35.1.2. Theorem. If a matrix A is normal then ρ(A)=kAk .
√ s
35.2. Theorem. kAk ≤kAk ≤ nkAk .
s e s
Theinvarianceofthematrixnormandsingularvalues.
A+A∗
35.3.1. Theorem. LetS beanHermitianmatrix. ThenkA− k
2
does not exceed kA−Sk, where k·k 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 kA−Uk ≤kA−Wk and if |A|=6 0, then the
e e
equality is only attained for W =U.
Problems
36. Schur’s complement and Hadamard’s product. Theorems of
Emily Haynsworth
CONTENTS 7
36.1.1. Theorem. If A>0 then (A|A11)>0.
36.1.4. Theorem. If Ak and Bk are the k-th principal submatrices of
positive definite order n matrices A and B, then
à ! à !
|A+B|≥|A| 1+nX−1|Bk| +|B| 1+nX−1|Ak| .
k=1|Ak| k=1|Bk|
Hadamard’sproductA◦B.
36.2.1. Theorem. If A>0 and B>0 then A◦B>0.
Oppenheim’sinequality
Problems
37. Nonnegative matrices
Wielandt’stheorem
Problems
38. Doubly stochastic matrices
Birkhoff’stheorem. H.Weyl’sinequality.
Solutions
Chapter VII. Matrices in algebra and calculus
39. Commuting matrices
ThespaceofsolutionsoftheequationAX=XAforX withthegivenA
ofordern.
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
40.2. Theorem. If trA = 0 then there exist matrices X and Y such
that [X,Y]=A and either (1) trY =0 and an Hermitian matrix X or (2)
X and Y have prescribed eigenvalues.
40.3. Theorem. Let A,B be matrices such that ads X = 0 implies
A
ads B=0 for some s>0. Then B=g(A) for a polynomial g.
X
40.4. Theorem. MatricesA1,...,An canbesimultaneouslytriangular-
izedoverCifandonlyifthematrixp(A1,...,An)[Ai,Aj]isanilpotentone
for any polynomial p(x1,...,xn) in noncommuting indeterminates.
40.5. Theorem. Ifrank[A,B]≤1,thenAandB canbesimultaneously
triangularized over C.
Problems
41. Quaternions and Cayley numbers. Clifford algebras
Isomorphisms so(3,R) ∼= su(2) and so(4,R) ∼= so(3,R)⊕so(3,R). The
vectorproductsinR3andR7. Hurwitz-Radonfamiliesofmatrices. Hurwitz-
Radon’numberρ(2c+4d(2a+1))=2c+8d.
41.7.1. Theorem. The identity of the form
(x2+···+x2)(y2+···+y2)=(z2+···+z2),
1 n 1 n 1 n
where zi(x,y) is a bilinear function, holds if and only if m≤ρ(n).
41.7.5. Theorem. In the space of real n×n matrices, a subspace of
invertible matrices of dimension m exists if and only if m≤ρ(n).
Other applications: algebras with norm, vector product, linear vector
fieldsonspheres.
CliffordalgebrasandCliffordmodules.
8
Problems
42. Representations of matrix algebras
Completereducibilityoffinite-dimensionalrepresentationsofMat(Vn).
Problems
43. The resultant
Sylvester’smatrix,Bezout’smatrixandBarnett’smatrix
Problems
44. The general inverse matrix. Matrix equations
44.3. Theorem. a)The equationAX−XA=C issolvable if and only
ţ ű ţ ű
A O A C
if the matrices and are similar.
O B O B
ţ ű
A O
b)The equation AX−YA=C is solvable if and only if rank
O B
ţ ű
A C
=rank .
O B
Problems
45. Hankel matrices and rational functions
46. Functions of matrices. Differentiation of matrices
DifferentialequationX˙ =AX andtheJacobiformulafordet A.
Problems
47. Lax pairs and integrable systems
48. Matrices with prescribed eigenvalues
48.1.2. Theorem. Foranypolynomialf(x)=xn+c1xn−1+···+cnand
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 x1,...,xn so that the eigenvalues
of A are given complex numbers; there are finitely many sets {x1,...,xn}
satisfying this condition.
Solutions
Appendix
Eisenstein’scriterion,Hilbert’sNullstellensats.
Bibliography
Index
CONTENTS 9
PREFACE
Thereareverymanybooksonlinearalgebra,amongthemmanyreallywonderful
ones (see e.g. the list of recommended literature). One might think that one does
not need any more books on this subject. Choosing one’s 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 of the known theorems. Besides, more than a few interesting old
results are ignored, so far, by text-books.
InthisbookItriedtocollectthemostattractiveproblemsandtheoremsoflinear
algebra still accessible to first year students majoring or minoring in mathematics.
Thecomputationalalgebrawasleftsomewhataside. Themajorpartofthebook
contains results known from journal publications only. I believe that they will be
of interest to many readers.
I assume that the reader is acquainted with main notions of linear algebra:
linear space, basis, linear map, the determinant of a matrix. Apart from that,
all the essential theorems of the standard course of linear algebra are given here
with complete proofs and some definitions from the above list of prerequisites is
recollected. I made 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 inside the book are natural: 36.2 means subsection 2 of sec. 36;
Problem 36.2 is Problem 2 from sec. 36; Theorem 36.2.2 stands for Theorem 2
from 36.2.
Acknowledgments. The book is based on a course I read at the Independent
UniversityofMoscow,1991/92. Iamthankfultotheparticipantsforcommentsand
to D. V. Beklemishev, D. B. Fuchs, A. I. Kostrikin, V. S. Retakh, A. N. Rudakov
and A. P. Veselov for fruitful discussions of the manuscript.
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