Page 338 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 31: Finite Dimensional Spectral Theory

This definition can be extended similarly for the case when A, B are operators on a Hilbert space. Notes

Notes
1. The matrices in A are similar iff they are the matrices of a single operator on H
n
relative to two different basis H.
2. Similar matrices have the same determinant.

31.1.3 Determinant of an Operator

Let T be an operator on an n-dimensional Hilbert space H. Then the determinant of the operator
T is the determinant of the matrix of T, namely [T] with respect to any ordered basis for H.

Following we given properties of a the determinant of an operator on a finite dimensional
Hilbert space H.
(i) det (I) = 1, I being identity operator.

Since det (I) = det ([I]) = det ([ ]) = 1.
ij
(ii) det (T T ) = (det T ) (det T )
1 2 1 2
(iii) det (T) 0 [T] is non-singular
det ([T]) 0.

Hence det (T) 0 [T] is non-singular.

31.1.4 Spectral Analysis

Definition: Eigenvalues

Let T be bounded linear operator on a Hilbert space H. Then a scalar is called an eigenvalue of
T if there exists a non-zero vector x in H such that Tx = x.
Eigenvalues are sometimes referred as characteristic values or proper values or spectral values.

Definition: Eigenvectors

If is an eigenvalue of T, then any non-zero vector x H such Tx = x, is called a eigenvector
(characteristic vector or proper vector or spectral vector) of T.

Properties of Eigenvalues and Eigenvectors

Notes If the Hilbert space has no non-zero vectors then T cannot have any eigenvectors
and consequently the whole theory reduces to triviality. So we shall assume that H 0
throughout this unit.

1. If x is an eigenvector of T corresponding to the eigenvalue and is a non-zero
scalar, then x is also an eigenvector of T corresponding to the same eigenvalue .
Contd...

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