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210 ALGEBRA

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

i
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)= m λ i .
i
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,

m
det A = λ i .
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 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.In the finite-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.Let A 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

(Ax) ⋅ x
λ m+1 =min max .
Y E m x⊥Y 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
k
form a = λ k .
k