Handbook of mathematics for engineers and scienteists part 35 docx

206 ALGEBRA
The rank of a linear operator A is the dimension of its range: rank (A)=dim(imA).
Properties of the rank of a linear operator:
rank (AB) ≤ min{rank (A), rank (B)},
rank (A)+rank(B)–n ≤ rank (AB),
where A and B are linear operators in L(V, V)andn =dimV.
Remark. If rank (A)=n then rank (AB)=rank(BA)=rank(B).
THEOREM.
Let
A : V→V
be a linear operator. Then the following statements are
equivalent:
1.
A
is invertible (i.e., there exists
A
–1
).
2.
ker A = 0
.
3.
im A = V
.
4.
rank (A)=dimV
.
5.6.1-5. Notion of a adjoint operator. Hermitian operators.
Let A L(V, V) be a bounded linear operator in a Hilbert space V. The operator A
∗
in

L(V, V) is called its adjoint operator if
(Ax) ⋅ y = x ⋅ (A
∗
y)
for all x and y in V.
T
HEOREM.
Any bounded linear operator
A
in a Hilbert space has a unique adjoint
operator.
Properties of adjoint operators:
(A + B)
∗
= A
∗
+ B
∗
,(λA)
∗
=
¯
λA
∗
,(A
∗
)
∗
= A,
(AB)

∗
= B
∗
A
∗
, O
∗
= O, I
∗
= I,
(A
–1
)
∗
=(A
∗
)
–1
, A
∗
 = A, A
∗
A = A
2
,
(Ax) ⋅ (By) ≡ x ⋅ (A
∗
By) ≡ (B
∗
Ax) ⋅ y for all x and y in V,

where A and B are bounded linear operators in a Hilbert space V,
¯
λ is the complex conjugate
of a number λ.
A linear operator A
L(V, V) in a Hilbert space V is said to be Hermitian (self-adjoint) if
A
∗
= A or (Ax) ⋅ y = x ⋅ (Ay).
A linear operator A
(V, V) in a Hilbert space V is said to be skew-Hermitian if
A
∗
=–A or (Ax) ⋅ y =–x ⋅ (Ay).
5.6.1-6. Unitary and normal operators.
A linear operator U L(V, V) in a Hilbert space V is called a unitary operator if for all x
and y in V, the following relation holds:
(Ux) ⋅ (Uy)=x ⋅ y.
This relation is called the unitarity condition.
5.6. LINEAR OPERATORS 207
Properties of a unitary operator U:
U
∗
= U
–1
or U
∗
U = UU
∗
= I,

Ux = x for all x in V.
A linear operator A in L(V, V)issaidtobenormal if
A
∗
A = AA
∗
.
T
HEOREM.
A bounded linear operator
A
is normal if and only if
Ax = Ax
.
Remark. Any unitary or Hermitian operator is normal.
5.6.1-7. Transpose, symmetric, and orthogonal operators.
The transpose operator of a bounded linear operator A L(V, V) in a real Hilbert space V
is the operator A
T
L(V, V) such that for all x, y in V, the following relation holds:
(Ax) ⋅ y = x ⋅ (A
T
y).
T
HEOREM.
Any bounded linear operator
A
in a real Hilbert space has a unique transpose
operator.
The properties of transpose operators in a real Hilbert space are similar to the properties

of adjoint operators considered in Paragraph 5.6.1-5 if one takes A
T
instead of A
∗
.
A linear operator A
L(V, V) in a real Hilbert space V is said to be symmetric if
A
T
= A or (Ax) ⋅ y = x ⋅ (Ay).
A linear operator A
L(V, V) in a real Hilbert space V is said to be skew-symmetric if
A
T
=–A or (Ax) ⋅ y =–x ⋅ (Ay).
The properties of symmetric linear operators in a real Hilbert space are similar to the
properties of Hermitian operators considered in Paragraph 5.6.1-5 if one takes A
T
instead
of A
∗
.
A linear operator P
L(V, V) in a real Hilbert space V is said to be orthogonal if for
any x and y in V, the following relations hold:
(Px) ⋅ (Py)=x ⋅ y.
This relation is called the orthogonality condition.
Properties of orthogonal operator P:
P
T

= P
–1
or P
T
P = PP
T
= I,
Px = x for all x in V.
5.6.1-8. Positive operators. Roots of an operator.
A Hermitian (symmetric, in the case of a real space) operator A is said to be
a) nonnegative (resp., nonpositive), and one writes A ≥ 0 (resp., A ≤ 0)if(Ax) ⋅ x ≥ 0
(resp., (Ax) ⋅ x ≤ 0)foranyx in V.
b) positive or positive definite (resp., negative or negative definite) and one writes A > 0
(A < 0)if(Ax) ⋅ x > 0 (resp., (Ax) ⋅ x < 0)foranyx ≠ 0.
An mth root of an operator A is an operator B such that B
m
= A.
T
HEOREM.
If
A
is a nonnegative Hermitian (symmetric) operator, then for any positive
integer
m
there exists a unique nonnegative Hermitian (symmetric) operator
A
1/m
.
208 ALGEBRA
5.6.1-9. Decomposition theorems.

THEOREM 1.
For any bounded linear operator
A
in a Hilbert space
V
, the operator
H
1
=
1
2
(A + A
∗
)
is Hermitian and the operator
H
2
=
1
2
(A – A
∗
)
is skew-Hermitian. The
representation of
A
as a sum of Hermitian and skew-Hermitian operators is unique:
A =
H
1

+ H
2
.
THEOREM 2.
For any bounded linear operator
A
in a real Hilbert space, the operator
S
1
=
1
2
(A + A
T
)
is symmetric and the operator
S
2
=
1
2
(A – A
T
)
is skew-symmetric. The
representation of
A
as a sum of symmetric and skew-symmetric operators is unique:
A =
S

1
+ S
2
.
THEOREM 3.
For any bounded linear operator
A
in a Hilbert space,
AA
∗
and
A
∗
A
are
nonnegative Hermitian operators.
THEOREM 4.
For any linear operator
A
in a Hilbert space
V
,thereexist
polar decom-
positions
A = QU
and
A = U
1
Q
1

,
where
Q
and
Q
1
are nonnegative Hermitian operators,
Q
2
= AA
∗
,
Q
2
1
= A
∗
A
,and
U
,
U
1
are unitary operators. The operators
Q
and
Q
1
are always unique, while the operators
U

and
U
1
are unique only if
A
is nondegenerate.
5.6.2. Linear Operators in Matrix Form
5.6.2-1. Matrices associated with linear operators.
Let A be a linear operator in an n-dimensional linear space V with a basis e
1
, , e
n
.Then
there is a matrix [a
j
j
] such that
Ae
j
=
n

i=1
a
i
j
e
i
.
The coordinates y

j
of the vector y = Ax in that basis can be represented in the form
y
i
=
n

j=1
a
i
j
x
j
(i = 1, 2, , n), (5.6.2.1)
where x
j
are the coordinates of x in the same basis e
1
, , e
n
. The matrix A ≡ [a
i
j
]ofsize
n × n is called the matrix of the linear operator A in a given basis e
1
, , e
n
.
Thus, given a basis e

1
, , e
n
, any linear operator y = Ax can be associated with its
matrix in that basis with the help of (5.6.2.1).
If A is the zero operator, then its matrix is the zero matrix in any basis. If A is the unit
operator, then its matrix is the unit matrix in any basis.
T
HEOREM 1.
Let
e
1
,

,
e
n
be a given basis in a linear space
V
and let
A ≡ [a
i
j
]
be a
given square matrix of size
n × n
. Then there exists a unique linear operator
A : V→V
whose matrix in that basis coincides with the matrix

A
.
THEOREM 2.
The rank of a linear operator
A
is equal to the rank of its matrix
A
in any
basis:
rank (A)=rank(A)
.
THEOREM 3.
A linear operator
A : V→V
is invertible if and only if
rank (A)=dimV
.
In this case, the matrix of the operator
A
is invertible.
5.6. LINEAR OPERATORS 209
5.6.2-2. Transformation of the matrix of a linear operator.
Suppose that the transition from the basis e
1
, , e
n
to anotherbasis

e
1

, ,

e
n
is determined
by a matrix U ≡ [u
ij
]ofsizen × n,i.e.

e
i
=
n

j=1
u
ij
e
j
(i = 1, 2, , n).
T
HEOREM.
Let
A
and

A
be the matrices of a linear operator
A
in the basis

e
1
,

,
e
n
and the basis

e
1
,

,

e
n
, respectively. Then
A = U
–1

AU
or

A = UAU
–1
.
Note that the determinant of the matrix of a linear operator does not depend on the
basis: det A =det


A. Therefore, one can correctly define the determinant det A of a linear
operator as the determinant of its matrix in any basis:
det A =detA.
The trace ofthe matrix of a linear operator, Tr(A), isalso independent of the basis. Therefore,
one can correctly define the trace Tr(A) of a linear operator as the trace of its matrix in any
basis:
Tr(A)=Tr(A).
In the case of an orthonormal basis, a Hermitian, skew-Hermitian, normal, or unitary
operator in a Hilbert space corresponds to a Hermitian, skew-Hermitian, normal, or unitary
matrix; and a symmetric, skew-symmetric, or transpose operator in a real Hilbert space
corresponds to a symmetric, skew-symmetric, or transpose matrix.
5.6.3. Eigenvectors and Eigenvalues of Linear Operators
5.6.3-1. Basic definitions.
1
◦
. A scalar λ is called an eigenvalue of a linear operator A in a vector space V if there is
a nonzero element x in V such that
Ax = λx.(5.6.3.1)
A nonzero element x for which (5.6.3.1) holds is called an eigenvector of the operator A
corresponding to the eigenvalue λ. Eigenvectors corresponding to distinct eigenvalues are
linearly independent. For an eigenvalue λ ≠ 0,theinverseμ = 1/λ is called a characteristic
value of the operator A.
T
HEOREM.
If
x
1
,

,

x
k
are eigenvectors of an operator
A
corresponding to its eigen-
value
λ
,then
α
1
x
1
+ ···+ α
k
x
k
(
α
2
1
+ ···+ α
2
k
≠ 0
) is also an eigenvector of the operator
A
corresponding to the eigenvalue
λ
.
The geometric multiplicity m

i
of an eigenvalue λ
i
is the maximal number of linearly
independent eigenvectors corresponding to the eigenvalue λ
i
. Thus, the geometric multi-
plicity of λ
i
is the dimension of the subspace formed by all eigenvectors corresponding to
the eigenvalue λ
i
.
The algebraic multiplicity m

i
of an eigenvalue λ
i
of an operator A is equal to the
algebraic multiplicity of λ
i
regarded as an eigenvalue of the corresponding matrix A.
210 ALGEBRA
The algebraic multiplicity m

i
of an eigenvalue λ
i
is always not less than the geometric
multiplicity m

i
of this eigenvalue.
The trace Tr(A) is equal to the sum of all eigenvalues of the operator A, each eigenvalue
counted according to its multiplicity, i.e.,
Tr(A)=

i
m

i
λ
i
.
The determinant det A is equal to the product of all eigenvalues of the operator A, each
eigenvalue entering the product according to its multiplicity,
det A =

i
λ
m

i
i
.
5.6.3-2. Eigenvectors and eigenvalues of normal and Hermitian operators.
Properties of eigenvalues and eigenvectors of a normal operator:
1. A normal operator A in a Hilbert space V and its adjoint operator A
∗
have the same
eigenvectors and their eigenvalues are complex conjugate.

2. For a normal operator A in aHilbert space V, there is a basis {e
k
} formed by eigenvectors
of the operators A and A
∗
. Therefore, there is a basis in V in which the operator A has
a diagonal matrix.
3. Eigenvectors corresponding to distinct eigenvalues of a normal operator are mutually
orthogonal.
4. Any bounded normal operator A in a Hilbert space V is reducible. The space V can
be represented as a direct sum of the subspace spanned by an orthonormal system of
eigenvectors of A and the subspace consisting of vectors orthogonal to all eigenvectors
of A.Inthefinite-dimensional case, an orthonormal system of eigenvectors of A is a
basis of V.
5. The algebraic multiplicity of any eigenvalue λ of a normal operator is equal to its
geometric multiplicity.
Properties of eigenvalues and eigenvectors of a Hermitian operator:
1. Since any Hermitian operator is normal, all properties of normal operators hold for
Hermitian operators.
2. All eigenvalues of a Hermitian operator are real.
3. Any Hermitian operator A in an n-dimensional unitary space has n mutually orthogonal
eigenvectors of unit length.
4. Any eigenvalue of a nonnegative (positive) operator is nonnegative (positive).
5. Minimax property.LetA be a Hermitian operator in an n-dimensional unitary space V,
and let E
m
be the set of all m-dimensional subspaces of V (m < n). Then the eigenvalues
λ
1
, , λ

n
of the operator A (λ
1
≥ ≥ λ
n
) can be defined by the formulas
λ
m+1
=min
Y E
m
max
x⊥Y
(Ax) ⋅ x
x ⋅ x
.
6. Let i
1
, , i
n
be an orthonormal basis in an n-dimensional space V, and let all i
k
are eigenvectors of a Hermitian operator A, i.e., Ai
k
= λ
k
i
k
. Then the matrix of
the operator A in the basis i

1
, , i
n
is diagonal and its diagonal elements have the
form a
k
k
= λ
k
.
5.6. LINEAR OPERATORS 211
7. Let i
1
, , i
n
be an arbitrary orthonormal basis in an n-dimensional Euclidean space V.
Then the matrix of an operator A in the basis i
1
, , i
n
is symmetric if and only if the
operator A is Hermitian.
8. In an orthonormal basis i
1
, , i
n
formed by eigenvectors of a nonnegative Hermitian
operator A, the matrix of the operator A
1/m
has the form

⎛
⎜
⎜
⎜
⎝
λ
1/m
1
0 ··· 0
0 λ
1/m
2
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
00··· λ
1/m
n
⎞
⎟

⎟
⎟
⎠
.
5.6.3-3. Characteristic polynomial of a linear operator.
Consider the finite-dimensional case. The algebraic equation
f
A
(λ) ≡ det(A – λI)=0 (5.6.3.2)
of degree n is called the characteristic equation of the operator A and f
A
(λ) is called the
characteristic polynomial of the operator A.
Since the value of the determinant det(A – λI) does not depend on the basis, the
coefficients of λ
k
(k = 0, 1, , n) in the characteristic polynomial f
A
(λ)areinvariants
(i.e., quantities whose values do not depend on the basis). In particular, the coefficient
of λ
k–1
is equal to the trace of the operator A.
In the finite-dimensional case, λ is an eigenvalue of a linear operator A if and only if λ is
a root of the characteristic equation (5.6.3.2) of the operator A. Therefore, a linear operator
always has eigenvalues.
In the case of a real space, a root of the characteristic equation can be an eigenvalue of
a linear operator only if this root is real. In this connection, it would be natural to find a
class of linear operators in a real Euclidean space for which all roots of the corresponding
characteristic equations are real.

T
HEOREM.
The matrix
A
of a linear operator
A
in a given basis
i
1
,

,
i
n
is diagonal if
and only if all
i
i
are eigenvectors of this operator.
5.6.3-4. Bounds for eigenvalues of linear operators.
The modulus of any eigenvalue λ of a linear operator A in an n-dimensional unitary space
satisfies the estimate:
|λ| ≤ min(M
1
, M
2
), M
1
=max
1≤i≤n

n

j=1
|a
ij
|, M
2
=max
1≤j≤n
n

i=1
|a
ij
|,
where A ≡ [a
ij
] is the matrix of the operator A. The real and the imaginary parts of
eigenvalues satisfy the estimates:
min
1≤i≤n
(Re a
ii
– P
i
) ≤ Re λ ≤ max
1≤i≤n
(Re a
ii
+ P

i
),
min
1≤i≤n
(Im a
ii
– P
i
) ≤ Im λ ≤ max
1≤i≤n
(Im a
ii
+ P
i
),
212 ALGEBRA
where P
i
=
n

j=1, j≠i
|a
ij
|,andP
i
can be replaced by Q
i
=
n


j=1, i≠i
|a
ji
|.
The modulus of any eigenvalue λ of a Hermitian operator A in an n-dimensional unitary
space satisfies the inequalities
|λ|
2
≤

i

j
|a
ij
|
2
, |λ| ≤ A =sup
x=1
[(Ax) ⋅ x],
and its smallest and its largest eigenvalues, denoted, respectively, by m and M, can be
found from the relations
m =inf
x=1
[(Ax) ⋅ x], M =sup
x=1
[(Ax) ⋅ x].
5.6.3-5. Spectral decomposition of Hermitian operators.
Let i

1
, , i
n
be a fixed orthonormal basis in an n-dimensional unitary space V.Thenany
element of V can be represented in the form (see Paragraph 5.4.2-2)
x =
n

j=1
(x ⋅ i
j
)i
j
.
The operator P
k
(k = 1, 2, , n)defined by
P
k
x =(x ⋅ i
k
)i
k
is called the projection onto the one-dimensional subspace generated by the vector i
k
.The
projection P
k
is a Hermitian operator.
Properties of the projection P

k
:
P
k
P
l
=

P
k
for k = l,
O for k ≠ l,
P
m
k
= P
k
(m = 1, 2, 3, ),
n

j=1
P
j
= I,whereI is the identity operator.
For a normal operator A, there is an orthonormal basis consisting of its eigenvectors,
Ai
k
= λi
k
. Then one obtains the spectral decomposition of a normal operator:

A
k
=
n

j=1
λ
k
j
P
j
(k = 1, 2, 3, ). (5.6.3.3)
Consider an arbitrary polynomial p(λ)=
m

j=1
c
j
λ
j
.Bydefinition, p(A)=
m

j=1
c
j
A
j
. Then,
using (5.6.3.3), we get

p(A)=
m

i=1
p(λ
i
)P
i
.
C
AYLEY-HAMILTON THEOREM.
Every normal operator satisfies its own characteristic
equation, i.e.,
f
A
(A)=O
.