Page 295 - DMTH502_LINEAR_ALGEBRA

Unit 26: Unitary Operators and Normal Operators

 With the help of a few theorems and examples the properties of unitary operators are Notes
explained.

 The distinction between unitary operators, orthogonal operators and normal operators is
established.
 With the help of a few theorem it is shown that for every normal matrix A, there is a
–1
unitary matrix P such that P AP is a diagonal matrix.
26.4 Keywords

General Linear Group: A general linear group denotes the set of all invertible complex n n
matrices and is denoted by GL(n).

Isomorphism: An isomorphism of inner product spaces V onto W is a vector space isomorphism
of the linear operator T of V onto W which also preserves inner products.
t
Orthogonal: A real or complex n n matrix A is said to be orthogonal if A A = I.
Unitary: A complex n n matrix A is called unitary if A*A = 1.
Unitary Operator: A unitary operator on an inner product space is isomorphism of the space
onto itself.

26.5 Review Questions

1 2 3
1. For 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.
2. If T is a normal operator. Prove that characteristic vectors for T which are associated with
distinct characteristic values are orthogonal.

26.6 Further Readings

Books Michael Artin Algebra

I N. Herstein Topics in Algebra
Kenneth Hoffman and Ray Kunze Linear Algebra

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