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 . . 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 =  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. 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