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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 another basis2 e 1 , ... ,2 e n is determined
by a matrix U ≡ [u ij ]ofsize n × n,i.e.

n

2 e i = u ij e j (i = 1, 2, ... , n).
j=1

THEOREM. Let A and A be the matrices of a linear operator A in the basis e 1 , ... , e n
2
and the basis 2 e 1 , ... , 2 e n , respectively. Then
–1
A = U AU or A = UAU .
–1 2
2
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
2
operator as the determinant of its matrix in any basis:
det A =det A.

The trace of the matrixof a linear operator, Tr(A), isalsoindependent 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,the inverse μ = 1/λ is called a characteristic
value of the operator A.
THEOREM. If x 1 , ... , x k are eigenvectors of an operator A corresponding to its eigen-
2
2
value λ,then α 1 x 1 + ··· + α k x k (α + ··· + α ≠ 0) is also an eigenvector of the operator A
1
k
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 of an eigenvalue λ i of an operator A is equal to the

i
algebraic multiplicity of λ i regarded as an eigenvalue of the corresponding matrix A.