4
Chapter IV. Matrices of special form
19. Symmetric and Hermitian matrices
Sylvester’s criterion. Sylvester’s law of inertia. Lagrange’s theorem on
quadratic forms. Courant-Fisher’s theorem.
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. An example of two Hermitian matrices which can not be simultane-
ously diagonalized. Simultaneous diagonalization of two semidefinite matri-
ces. Simultaneous diagonalization 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
21.1.1. Theorem. If A is a skew-symmetric matrix then A
2
≤ 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. Any skew-symmetric bilinear form can be expressed as
r
P
k=1
(x
2k−1
y
2k
− x
2k
y
2k−1
).
Problems
22. Orthogonal matrices. The Cayley transformation
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
23.1.1. Theorem. If an operator A is normal then Ker A
∗
= Ker A and
Im A
∗
= Im A.
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. Let A be an n ×n matrix. The matrix A is nilpotent
if and only if tr (A
p
) = 0 for each p = 1, . . . , n.
Nilpotent matrices and Young tableaux.
Problems
25. Projections. Idempotent matrices
25.2.1&2. Theorem. An idempotent operator P is an Hermitian one
if and only if a) Ker P ⊥ Im P ; or b) |Px| ≤ |x| for every x.
25.2.3. Theorem. Let P
1
, . . . , P
n
be Hermitian, idempotent operators.
The operator P = P
1
+ ···+P
n
is an idempotent one if and only if P
i
P
j
= 0
whenever i = j.
25.4.1. Theorem. Let V
1
⊕ ··· ⊕ V
k
, P
i
: V −→ V
i
be Hermitian
idempotent operators, A = P
1
+ ···+P
k
. Then 0 < det A ≤ 1 and det A = 1
if and only if V
i
⊥ V
j
whenever i = 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
equations AX − XB = C and AX − XB = λX.
Problems
28. Symmetric and skew-symmetric tensors
The Grassmann algebra. Certain canonical isomorphisms. Applications
of Grassmann algebra: proofs of Binet-Cauchy’s formula and Sylvester’s iden-
tity.
28.5.4. Theorem. Let Λ
B
(t) = 1 +
n
P
q=1
tr(Λ
q
B
)t
q
and S
B
(t) = 1 +
n
P
q=1
tr (S
q
B
)t
q
. Then S
B
(t) = (Λ
B
(−t))
−1
.
Problems
29. The Pfaffian
The Pfaffian of principal submatrices of the matrix M =
ř
ř
m
ij
ř
ř
2n
1
, where
m
ij
= (−1)
i+j+1
.
29.2.2. Theorem. Given a skew-symmetric matrix A we have
Pf (A + λ
2
M) =
n
X
k=0
λ
2k
p
k
, where p
k
=
X
σ
A
Ã
σ
1
. . . σ
2(n−k)
σ
1
. . . σ
2(n−k)
!
Problems
30. Decomposable skew-symmetric and symmetric tensors
30.1.1. Theorem. x
1
∧ ··· ∧ x
k
= y
1
∧ ··· ∧ y
k
= 0 if and only if
Span(x
1
, . . . , x
k
) = Span(y
1
, . . . , y
k
).
30.1.2. Theorem. S(x
1
⊗ ··· ⊗ x
k
) = S(y
1
⊗ ··· ⊗ y
k
) = 0 if and only
if Span(x
1
, . . . , x
k
) = Span(y
1
, . . . , y
k
).
Plu¨cker relations.
Problems
31. The tensor rank
Strassen’s algorithm. The set of all tensors of rank ≤ 2 is not closed. The
rank over R is not equal, generally, to the rank over C.
Problems
32. Linear transformations of tensor products
A complete description of the following types of transformations of
V
m
⊗ (V
∗
)
n
∼
=
M
m,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
−1
x, x) = max
y
(2(x, y) − (Ay, y)).
33.2.1. Theorem. Suppose A =
ţ
A
1
B
B
∗
A
2
ű
> 0. Then |A| ≤ |A
1
| ·
|A
2
|.
Hadamard’s inequality and Szasz’s inequality.
33.3.1. Theorem. Suppose α
i
> 0,
n
P
i=1
α
i
= 1 and A
i
> 0. Then
|α
1
A
1
+ ··· + α
k
A
k
| ≥ |A
1
|
α
1
+ ··· + |A
k
|
α
k
.
33.3.2. Theorem. Suppose A
i
≥ 0, α
i
∈ C. Then
|det(α
1
A
1
+ ··· + α
k
A
k
)| ≤ det(|α
1
|A
1
+ ··· + |α
k
|A
k
).
Problems
34. Inequalities for eigenvalues
Schur’s inequality. Weyl’s inequality (for eigenvalues of A + B).
34.2.2. Theorem. Let A =
ţ
B C
C
∗
B
ű
> 0 be an Hermitian matrix,
α
1
≤ ··· ≤ α
n
and β
1
≤ ··· ≤ β
m
the eigenvalues of A and B, respectively.
Then α
i
≤ β
i
≤ α
n+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 λ
i
and µ
i
be the eigenvalues of A and AA∗,
respectively; let σ
i
=
√
µ
i
. Let |λ
1
≤ ··· ≤ λ
n
, where n is the order of A.
Then |λ
1
. . . λ
m
| ≤ σ
1
. . . σ
m
.
34.4.2.Theorem. Let σ
1
≥ ··· ≥ σ
n
and τ
1
≥ ··· ≥ τ
n
be the singular
values of A and B. Then |tr (AB)| ≤
P
σ
i
τ
i
.
Problems
35. Inequalities for matrix norms
The spectral norm A
s
and the Euclidean norm A
e
, the spectral radius
ρ(A).
35.1.2. Theorem. If a matrix A is normal then ρ(A) = A
s
.
35.2. Theorem. A
s
≤ A
e
≤
√
nA
s
.
The invariance of the matrix norm and singular values.
35.3.1. Theorem. Let S be an Hermitian matrix. Then A −
A + A
∗
2
does not exceed A − S, where · is the Euclidean or operator norm.
35.3.2. Theorem. Let A = U S be the polar decomposition of A and
W a unitary matrix. Then A − U
e
≤ A −W
e
and if |A| = 0, then the
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|A
11
) > 0.
36.1.4. Theorem. If A
k
and B
k
are the k-th principal submatrices of
positive definite order n matrices A and B, then
|A + B| ≥ |A|
Ã
1 +
n−1
X
k=1
|B
k
|
|A
k
|
!
+ |B|
Ã
1 +
n−1
X
k=1
|A
k
|
|B
k
|
!
.
Hadamard’s product A ◦ B.
36.2.1. Theorem. If A > 0 and B > 0 then A ◦B > 0.
Oppenheim’s inequality
Problems
37. Nonnegative matrices
Wielandt’s theorem
Problems
38. Doubly stochastic matrices
Birkhoff’s theorem. H.Weyl’s inequality.
Solutions
Chapter VII. Matrices in algebra and calculus
39. Commuting matrices
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
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
s
A
X = 0 implies
ad
s
X
B = 0 for some s > 0. Then B = g(A) for a polynomial g.
40.4. Theorem. Matrices A
1
, . . . , A
n
can be simultaneously triangular-
ized over C if and only if the matrix p(A
1
, . . . , A
n
)[A
i
, A
j
] is a nilpotent one
for any polynomial p(x
1
, . . . , x
n
) in noncommuting indeterminates.
40.5. Theorem. If rank[A, B] ≤ 1, then A and B can be simultaneously
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
vector products in R
3
and R
7
. Hurwitz-Radon families of matrices. Hurwitz-
Radon’ number ρ(2
c+4d
(2a + 1)) = 2
c
+ 8d.
41.7.1. Theorem. The identity of the form
(x
2
1
+ ··· + x
2
n
)(y
2
1
+ ··· + y
2
n
) = (z
2
1
+ ··· + z
2
n
),
where z
i
(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
fields on spheres.
Clifford algebras and Clifford modules.
8
Problems
42. Representations of matrix algebras
Complete reducibility of finite-dimensional representations of Mat(V
n
).
Problems
43. The resultant
Sylvester’s matrix, Bezout’s matrix and Barnett’s matrix
Problems
44. The general inverse matrix. Matrix equations
44.3. Theorem. a) The equation AX − XA = C is solvable if and only
if the matrices
ţ
A O
O B
ű
and
ţ
A C
O B
ű
are similar.
b) The equation AX − Y A = C is solvable if and only if rank
ţ
A O
O B
ű
= rank
ţ
A C
O B
ű
.
Problems
45. Hankel matrices and rational functions
46. Functions of matrices. Differentiation of matrices
Differential equation
˙
X = AX and the Jacobi formula for det A.
Problems
47. Lax pairs and integrable systems
48. Matrices with prescribed eigenvalues
48.1.2. Theorem. For any polynomial f (x) = x
n
+c
1
x
n−1
+···+c
n
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
1
, . . . , x
n
so that the eigenvalues
of A are given complex numbers; there are finitely many sets {x
1
, . . . , x
n
}
satisfying this condition.
Solutions
Appendix
Eisenstein’s criterion, Hilbert’s Nullstellensats.
Bibliography
Index
CONTENTS 9
PREFACE
There are very many books on linear algebra, among them many really wonderful
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-bo oks.
In this book I tried to collect the most attractive problems and theorems of linear
algebra still accessible to first year students majoring or minoring in mathematics.
The computational algebra was left somewhat aside. The major part of the book
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
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 discussions of the manuscript.
Typeset by A
M
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10 PREFACE
Main notations and conventions
A =
a
11
. . . a
1n
. . . . . . . . .
a
m1
. . . a
mn
denotes a matrix of size m × n; we say that a square
n × n matrix is of order n;
a
ij
, sometimes denoted by a
i,j
for clarity, is the element or the entry from the
intersection of the i-th row and the j-th column;
(a
ij
) is another notation for the matrix A;
a
ij
n
p
still another notation for the matrix (a
ij
), where p ≤ i, j ≤ n;
det(A), |A| and det(a
ij
) all denote the determinant of the matrix A;
|a
ij
|
n
p
is the determinant of the matrix
a
ij
n
p
;
E
ij
— the (i, j)-th matrix unit — the matrix whose only nonzero element is
equal to 1 and occupies the (i, j)-th position;
AB — the product of a matrix A of size p × n by a matrix B of size n × q —
is the matrix (c
ij
) of size p ×q, where c
ik
=
n
j=1
a
ij
b
jk
, is the scalar product of the
i-th row of the matrix A by the k-th column of the matrix B;
diag(λ
1
, . . . , λ
n
) is the diagonal matrix of size n ×n with elements a
ii
= λ
i
and
zero offdiagonal elements;
I = diag(1, . . . , 1) is the unit matrix; when its size, n ×n, is needed explicitly we
denote the matrix by I
n
;
the matrix aI, where a is a numb er, is called a scalar matrix;
A
T
is the transposed of A, A
T
= (a
ij
), where a
ij
= a
ji
;
¯
A = (a
ij
), where a
ij
= a
ij
;
A
∗
=
¯
A
T
;
σ =
1 n
k
1
k
n
is a p ermutation: σ(i) = k
i
; the permutation
1 n
k
1
k
n
is often
abbreviated to (k
1
. . . k
n
);
sign σ = ( −1)
σ
=
1 if σ is even
−1 if σ is odd
;
Span(e
1
, . . . , e
n
) is the linear space spanned by the vectors e
1
, . . . , e
n
;
Given bases e
1
, . . . , e
n
and ε
ε
ε
1
, . . . ,ε
ε
ε
m
in spaces V
n
and W
m
, respectively, we
assign to a matrix A the operator A : V
n
−→ W
m
which sends the vector
x
1
.
.
.
x
n
into the vector
y
1
.
.
.
y
m
=
a
11
. . . a
1n
.
.
. . . .
.
.
.
a
m1
. . . a
mn
x
1
.
.
.
x
n
.
Since y
i
=
n
j=1
a
ij
x
j
, then
A(
n
j=1
x
j
e
j
) =
m
i=1
n
j=1
a
ij
x
j
ε
ε
ε
i
;
in particular, Ae
j
=
i
a
ij
ε
ε
ε
i
;
in the whole book except for §37 the notation
MAIN NOTATIONS AND CONVENTIONS 11
A > 0, A ≥ 0, A < 0 or A ≤ 0 denote that a real symmetric or Hermitian matrix
A is positive definite, nonnegative definite, negative definite or nonp ositive definite,
respectively; A > B means that A −B > 0; whereas in §37 they mean that a
ij
> 0
for all i, j, etc.
Card M is the cardinality of the set M, i.e, the number of elements of M;
A|
W
denotes the restriction of the operator A : V −→ V onto the subspace
W ⊂ V ;
sup the least upper bound (supremum);
Z, Q, R, C, H, O denote, as usual, the sets of all integer, rational, real, complex,
quaternion and octonion numbers, respectively;
N denotes the set of all positive integers (without 0);
δ
ij
=
1 if i = j,
0 otherwise.
12 PREFACECHAPTER I
DETERMINANTS
The notion of a determinant appeared at the end of 17th century in works of
Leibniz (1646–1716) and a Japanese mathematician, Seki Kova, also known as
Takakazu (1642–1708). Leibniz did not publish the results of his studies related
with 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 of the equations. In modern terms
he actually wrote about the indices i, j in the expression x
i
=
j
a
ij
y
j
.
Seki arrived at the notion of a determinant while solving the problem of 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.
He searched for multiple roots of a polynomial f(x) as common roots of f(x) and
f
(x). To find common roots of polynomials f(x) and g(x) (for f and g of small
degrees) Seki got determinant expressions. The main treatise by Seki was published
in 1674; there applications of the method are published, rather than the method
itself. He kept the main method in secret confiding only in his closest pupils.
In Europe, the first publication related to determinants, due to Cramer, ap-
peared 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 were proved only ad hoc when needed to
solve some other problem. Therefore, the theory of determinants had been develop-
ing slowly, left behind out of proportion as compared with the general development
of mathematics. A systematic presentation of the theory of determinants is mainly
associated with the names of Cauchy (1789–1857) and Jacobi (1804–1851).
1. Basic properties of determinants
The determinant of a square matrix A =
a
ij
n
1
is the alternated sum
σ
(−1)
σ
a
1σ(1)
a
2σ (2)
. . . a
nσ ( n)
,
where the summation is over all permutations σ ∈ S
n
. The determinant of the
matrix A =
a
ij
n
1
is denoted by det A or |a
ij
|
n
1
. If det A = 0, then A is called
invertible or nonsingular.
The following properties are often used to compute determinants. The reader
can easily verify (or recall) them.
1. Under the permutation of two rows of a matrix A its determinant changes
the sign. In particular, if two rows of the matrix are identical, det A = 0.
Typeset by A
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1. BASIC PROPERTIES OF DETERMINANTS 13
2. If A and B are square matrices, det
A C
0 B
= det A · det B.
3. | a
ij
|
n
1
=
n
j=1
(−1)
i+j
a
ij
M
ij
, where M
ij
is the determinant of the matrix
obtained from A by crossing out the ith row and the jth column of A (the row
(echelon) expansion of the determinant or, more precisely, the expansion with respect
to the ith row).
(To prove this formula one has to group the factors of a
ij
, where j = 1, . . . , n,
for a fixed i.)
4.
λα
1
+ µβ
1
a
12
. . . a
1n
.
.
.
.
.
. ···
.
.
.
λα
n
+ µβ
n
a
n2
. . . a
nn
= λ
α
1
a
12
. . . a
1n
.
.
.
.
.
. ···
.
.
.
α
n
a
n2
. . . a
nn
+ µ
β
1
a
12
. . . a
1n
.
.
.
.
.
. ···
.
.
.
β
n
a
n2
. . . a
nn
.
5. det(AB) = det A det B.
6. det(A
T
) = det A.
1.1. Before we start computing determinants, let us prove Cramer’s rule. It
appeared already in the first published paper on determinants.
Theorem (Cramer’s rule). Consider a system of linear equations
x
1
a
i1
+ ··· + x
n
a
in
= b
i
(i = 1, . . . , n),
i.e.,
x
1
A
1
+ ··· + x
n
A
n
= B,
where A
j
is the jth column of the matrix A =
a
ij
n
1
. Then
x
i
det(A
1
, . . . , A
n
) = det (A
1
, . . . , B, . . . , A
n
) ,
where the column B is inserted instead of A
i
.
Proof. Since for j = i the determinant of the matrix det(A
1
, . . . , A
j
, . . . , A
n
),
a matrix with two identical columns, vanishes,
det(A
1
, . . . , B, . . . , A
n
) = det (A
1
, . . . ,
x
j
A
j
, . . . , A
n
)
=
x
j
det(A
1
, . . . , A
j
, . . . , A
n
) = x
i
det(A
1
, . . . , A
n
).
If det(A
1
, . . . , A
n
) = 0 the formula obtained can be used to find solutions of a
system of linear equations.
1.2. One of the most often encountered determinants is the Vandermonde de-
terminant, i.e., the determinant of the Vandermonde matrix
V (x
1
, . . . , x
n
) =
1 x
1
x
2
1
. . . x
n−1
1
.
.
.
.
.
.
.
.
. ···
.
.
.
1 x
n
x
2
n
. . . x
n−1
n
=
i>j
(x
i
− x
j
).
To compute this determinant, let us subtract the (k − 1)-st column multiplied
by x
1
from the kth one for k = n, n − 1, . . . , 2. The first row takes the form
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