Page 294 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Therefore, A = 0 for every j > 1. In particular, A = 0, and since A is upper-triangular, it follows
1j 12
that
T = A .
2 22 2
Thus T* = A and A = 0 for all j 2. Continuing in this fashion, we find that A is diagonal.
2 22 2 2j
Theorem 12: Let V be a finite-dimensional complex inner product space and let T be any linear
operator on V. Then there is an orthonormal basis for V in which the matrix of T is upper
triangular.

Proof: Let n be the dimension of V. The theorem is true when n = 1, and we proceed by induction
on n, assuming the result is true for linear operators on complex inner product spaces of dimension
n – 1. Since V is a finite-dimensional complex inner product space, there is a unit vector in V
and a scalar c such that
T* = c .
Let W be the orthogonal complement of the subspace spanned by and let S be the restriction of
T to W. By Theorem 10, W is invariant under T. Thus S is a linear operator on W. Since W has
dimension n – 1, our inductive assumption implies the existence of an orthonormal basis { , . .
1
., } for W in which the matrix of S is upper-triangular; let = . Then { , . . ., } is an
n–1 n 1 n
orthonormal basis of V in which the matrix of T is upper-triangular.
This theorem implies the following result for matrices.
–1
Corollary: For every complex n n matrix A there is unitary matrix U such that U AU is upper-
triangular.
Now combining Theorem 12 and Theorem 11, we immediately obtain the following analogue
of Theorem 9 for normal operators.
Theorem 13: Let V be a finite-dimensional complex inner product space and T a normal operator
on V. Then V has an orthonormal basis consisting of characteristic vectors for T.
–1
Also for every normal matrix A, there is a unitary matrix P such that P AP is a diagonal matrix.
Self Assessment

3. For each of the following real symmetric matrices A, find a real orthogonal matrix P such
that P AP is diagonal
–1
1 1
(i) A =
1 1

4/3 2 /3
(ii) A =
2 /3 5/3

0 1
(iii) A =
1 0
4. Prove that T is normal if T = T + i T , where T and T are self-adjoint operators which
1 2 1 2
commute.
26.3 Summary

 In this unit we have studied unitary operators and normal operators.

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