Page 337 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 31.1 Finite Dimensional Spectral Theory

31.1.1 Linear Operators and Matrices on a Finite Dimensional Hilbert
Space

Let H be the given Hilbert space of dimension n with ordered basis B = {e , e , …, e } where the
1 2 n
ordered of the vector is taken into consideration. Let T (H) (the set of all bounded linear
operators). Since each vector in H is uniquely expressed as linear combination of the basis, we
n
can express Te as Te = e , where the n-scalars , , … are uniquely determined by Te .
j j ij i 1j 2j nj j
i 1
2
Then vectors Te , Te , …, Te determine uniquely the n scalars , i, j = 1, 2, … n. These n scalars
2
1 2 j ij
th
th
determine matrix with ( , , …, ) as the i row and ( , , … ) as its j column. We
i1 i2 in 1j 2j nj
denote this matrix by {T} and call this matrix as the matrix of the operator T with respect to the
ordered basis B.

11 12 1n

Hence  = [ ] = 21 22 2n
ij 

n1 n2 nn
We note that
(i) [0] = 0, which is the zero matrix.
(ii) [I] = I = [ ], which is a unit matrix of order n. Here is the Kronecker delta.
ij ij
Definition: The set of all n × n matrices denoted by A is complex algebra with respect to
n
addition, scalar multiplication and multiplication defined for matrices.
This algebra is called the total matrices algebra of order n.
Theorem 1: Let B be an ordered basis for a Hilbert space of dimension n. Let T (H) with (T) =
–1
–1
[ ], then T is singular [ ] is non-singular and we have [ ] = [T ].
ij ij ij
–1
Proof: T is non-singular iff there exists an operator T on H such that
–1
–1
T T = T T = I … (1)
–1
Since there is one-to-one correspondence between T and [T ],
–1
–1
(1) is true [T T] = [TT ] = [I]
–1
–1
from (2) [T ] [T] = [T] [T ] = [I] = [ ]
ij
–1
–1
so that [T ] [ ] = [ ] [T ] = [ ], [T] = [ ].
ij ij ij ij
–1
–1
[ ] is a non-singular and [ ] = [T ].
ij ij
This completes the proof of the theorem.
31.1.2 Similar Matrices
Let A, B are square matrix of order n over the field of complex number. Then B is said to be
similar to A if there exists a n × n non-singular matrix C over the field of complex numbers such
that
B = C AC.
–1

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