Handbook of mathematics for engineers and scienteists part 35 docx

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 adj

The rank of a linear operator A is the dimension of its range: rank (A) = dim (im A).

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) and n = dim V.

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 inV.

THEOREM 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 = A2,

(Ax)⋅(By)≡x⋅(A∗By)≡(B∗Ax)⋅y for all x and y inV,

where A and B are bounded linear operators in a Hilbert spaceV, ¯λ is the complex conjugate

of a number λ.

A linear operator AL( 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 UL( V, V) in a Hilbert space V is called a unitary operator if for all x

and y inV, the following relation holds:

(Ux)⋅(Uy) = x⋅y.

This relation is called the unitarity condition.

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) is said to be normal if

A∗A = AA∗ THEOREM A bounded linear operator A is normal if and only ifAx = 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 AL( V, V) in a real Hilbert space V

is the operator AT L( V, V) such that for all x, y in V, the following relation holds:

(Ax)⋅y = x⋅(ATy).

THEOREM 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 AT instead of A∗

A linear operator AL(V, V) in a real Hilbert space V is said to be symmetric if

AT = A or (Ax)⋅y = x⋅(Ay).

A linear operator AL( V, V) in a real Hilbert space V is said to be skew-symmetric if

AT = –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 AT instead

of A∗

A linear operator PL( V, V) in a real Hilbert space V is said to be orthogonal if for

any x and y inV, the following relations hold:

(Px)⋅(Py) = x⋅y.

This relation is called the orthogonality condition.

Properties of orthogonal operator P:

PT = P–1 or PTP = PPT = 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) for any x inV.

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) for any x≠ 0

An mth root of an operator A is an operator B such that B m= A.

THEOREM If A is a nonnegative Hermitian (symmetric) operator, then for any positive

integer m there exists a unique nonnegative Hermitian (symmetric) operator A1/m

5.6.1-9 Decomposition theorems.

THEOREM1 For any bounded linear operator A in a Hilbert space V, the operator

H1 = 12(A + A∗)is Hermitian and the operator H2 = 12(A – A∗)is skew-Hermitian The

representation of A as a sum of Hermitian and skew-Hermitian operators is unique: A =

H1+ H2

THEOREM2 For any bounded linear operator A in a real Hilbert space, the operator

S1 = 12(A + AT) is symmetric and the operator S2 = 12(A – AT) is skew-symmetric The

representation of A as a sum of symmetric and skew-symmetric operators is unique: A =

S1+ S2

THEOREM3 For any bounded linear operator A in a Hilbert space, AA∗and A∗Aare nonnegative Hermitian operators

THEOREM4 For any linear operator A in a Hilbert spaceV, there exist polar

decom-positions

A = QU and A = U1Q1,

where Q and Q1are nonnegative Hermitian operators, Q2 = AA∗, Q21 = A∗A, and U, U1 are unitary operators The operators Q and Q1 are always unique, while the operators U and U1are 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 e1 , , e n Then

there is a matrix [a j j] such that

Aej =

n



i=1

a i

jei

The coordinates y jof 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 e1, , e n The matrix A≡[a i j] of size

n×n is called the matrix of the linear operator A in a given basis e1, , e n

Thus, given a basis e1, , 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

THEOREM1 Let e1, , enbe a given basis in a linear spaceV 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.

THEOREM2 The rank of a linear operator A is equal to the rank of its matrix A in any basis: rank (A) = rank (A).

THEOREM3 A linear operator A :V → V is invertible if and only if rank (A) = dim V

In this case, the matrix of the operator A is invertible.

5.6.2-2 Transformation of the matrix of a linear operator.

Suppose that the transition from the basis e1, , e nto another basis2e1, ,2enis determined

by a matrix U ≡[u ij ] of size n×n, i.e.

2ei =

n



j=1

u ijej (i =1, 2, , n).

THEOREM Let A and 2 Abe the matrices of a linear operator A in the basis e1, , e n

and the basis2e1, ,2en, respectively Then

A = U–1AU2 or A2 = U AU– 1. Note that the determinant of the matrix of a linear operator does not depend on the

basis: det A = det 2 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 = det A.

The trace of the matrix of a linear operator, Tr(A), is also 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 inV such that

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, the inverse μ =1/λ is called a characteristic

value of the operator A.

THEOREM If x1, , x kare eigenvectors of an operator A corresponding to its

eigen-value λ, then α1x1+· · · + α kxk (α21+· · · + α2k≠ 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 iis 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.

The algebraic multiplicity m  i of an eigenvalue λ i is always not less than the geometric

multiplicity m iof 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 a Hilbert spaceV, there is a basis{e }formed by eigenvectors

of the operators A and A∗ Therefore, there is a basis inV 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 In the finite-dimensional case, an orthonormal system of eigenvectors of A is a

basis ofV.

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 Let A be a Hermitian operator in an n-dimensional unitary space V,

and letE m be the set of all m-dimensional subspaces of V (m < n) Then the eigenvalues

λ1, , λ n of the operator A (λ1 ≥ .≥λ n) can be defined by the formulas

λ m+1 = min

Y E m

max

x⊥Y

(Ax)⋅x

x⋅x .

6 Let i1, , 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., Aik = λ kik Then the matrix of

the operator A in the basis i1, , i n is diagonal and its diagonal elements have the

form a k k = λ k

7 Let i1, , i n be an arbitrary orthonormal basis in an n-dimensional Euclidean space V.

Then the matrix of an operator A in the basis i1, , i nis symmetric if and only if the

operator A is Hermitian.

8 In an orthonormal basis i1, , i nformed by eigenvectors of a nonnegative Hermitian

operator A, the matrix of the operator A1/mhas the form

⎛

⎜

⎜

⎝

λ1/m

0 λ1/m

2 · · · 0

. .

0 0 · · · λ1n /m

⎞

⎟

⎟

⎠.

5.6.3-3 Characteristic polynomial of a linear operator

Consider the finite-dimensional case The algebraic equation

fA(λ)≡det(A – λI) =0 (5.6.3.2)

of degree n is called the characteristic equation of the operator A and fA(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 fA(λ) are invariants

(i.e., quantities whose values do not depend on the basis) In particular, the coefficient

of λ k–1is 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

THEOREM The matrix A of a linear operator A in a given basis i1, , i nis diagonal if

and only if all iiare 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(M1, M2), M1= max

1≤i≤n

n



j=1

|a ij|, M2 = 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),

where P i =

n



j=1 ,j≠i

|a ij|, and P 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 i1, , i n be a fixed orthonormal basis in an n-dimensional unitary space V Then any

element ofV can be represented in the form (see Paragraph 5.4.2-2)

x =

n



j=1

(x⋅i )ij

The operator Pk (k =1,2, , n) defined by

Pkx = (x⋅ik)ik

is called the projection onto the one-dimensional subspace generated by the vector i k The

projection Pkis a Hermitian operator

Properties of the projection Pk:

PkPl=P

k for k = l,

O for k≠l, Pm k = Pk (m =1,2,3, ),

n



j=1Pj = I, where I is the identity operator.

For a normal operator A, there is an orthonormal basis consisting of its eigenvectors,

Aik = λi k Then one obtains the spectral decomposition of a normal operator:

Ak=

n



j=1

λ k

jPj (k =1, 2, 3, ). (5.6.3.3)

Consider an arbitrary polynomial p(λ) =m

j=1c j λ

j By definition, p(A) =m

j=1c jA

j Then,

using (5.6.3.3), we get

p(A) =

m



i=1

p(λ i)Pi

CAYLEY-HAMILTON THEOREM Every normal operator satisfies its own characteristic

equation, i.e., fA (A) = O.