Page 329 - DMTH502_LINEAR_ALGEBRA

Unit 29: Spectral Theory and Properties of Normal Operators

equivalent to T . Thus for each j there are orthonormal bases B and B of W and W , respectively Notes
j j j j j
such that
T
[ ] [ ] .
T
j B j j B j
Now let U be the linear transformation of V into V that maps each B onto B . Then U is a unitary
j j
transformation of V onto V such that UTU = T .
–1
Self Assessment
1. If U and T are normal operators which commute, prove that U + T and UT are normal.
A
2. Let A be an n × n matrix with complex entries such that A* = –A and let B = e . Show that
tr A
(a) det B = e ;
(b) B* = e ;
–A
(c) B is unitary.
3. For

1 2 3
A = 2 3 4
3 4 5

–1
there is a real orthogonal matrix p such that P AP = D is diagonal. Find such a diagonal
matrix D.
29.3 Summary

 The properties of unitary operators, normal operators or self-adjoint operators are studied
further. This study is an improvement of the results of unit 26.

 It is seen that a diagonalizable normal operator T on a finite dimensional inner product
space is either a self-adjoint, non-negative or unitary according as each characteristic
value of T is real, non-negative or of absolute value 1.

 If A is a normal matrix with real (complex) entries, then there is a real orthogonal (unitary)
–1
matrix P such that P AP is in rational canonical form.
29.4 Keywords

A Unitary Transformation: Let V and V’ be inner product spaces over the same field. A linear
transformation U: V V’ is called a unitary transformation if it preserves inner product.
Polar Decomposition: We call T = UN a polar decomposition for T on a finite dimensional inner
product space where U is a unitary operator and a unique non-negative linear operator on V.
The Non-negative: The non-negative operator T on an inner product space is self-adjoint and
(T | ) 0 for every in V.
The Spectral Resolution: The decomposition of the linear operator T as the sum of orthogonal
projections, i.e.

k
T C E .
i i
i 1

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