Page 289 - DMTH502_LINEAR_ALGEBRA

Unit 26: Unitary Operators and Normal Operators

Let GL(n) denote the set of all invertible complex n × n matrices. Then GL(n) is also a group Notes
under matrix multiplication. This group is called the general linear group. Theorem 5 is
equivalent to the following result.
+
Corollary: For each B in GL(n) there exist unique matrices N and U such that N is in T (n), U is in
U(n), and
B = N . U.
Proof: By the theorem there is a unique matrix M in T (n) such that MB is in U(n). Let MB = U and
+
+
–1
N = M . Then N is in T (n) and B = N . U. On the other hand, if we are given any elements N and
U such that N is in T (n), U is in U(n), and B = N . U, then N B is in U(n) and N is the unique
–1
–1
+
matrix M which is characterized by the theorem; furthermore U is necessarily N B.
–1
2
Example 7: Let x and x be real numbers such that x 2 x = 1 and x 0. Let
1 2 1 2 1
x x 0
1 2
B = 0 1 0 .
0 0 1
Applying the Gram-Schmidt process to the rows of B, we obtain the vectors
= (x , x , 0)
1 1 2
= (0, 1, 0) – x (x , x , 0)
2 2 1 2
= x (– x , x , 0)
1 2 1
= (0, 0, 1).
3
Let U be the matrix with rows , ( /x ), . Then U is unitary, and
1 2 1 3
1 0 0
x 1 x 2 0 x 1 x x 2 0
1
U = x 2 x 1 0 2 0 0 1 0
0 0 1 x 1 x 1 0 0 1
0 0 1

Now multiplying by the inverse of

1 0 0
x 2 1
M = 0
x 1 x 1
0 0 1

we find that

x x 2 0 1 0 0 x 1 x 2 0
1
0 1 0 = x x 1 0 x 2 x 1 0
2
0 0 1 0 0 1 0 0 1
Let us now consider briefly change of coordinates in an inner product space. Suppose V is a
finite-dimensional inner product space and that = { , …, } and = { ,, …, } are two
1 n 1 n
ordered orthonormal bases for V. There is a unique (necessarily invertible) n × n matrix P such
that

[ ] = P [ ]
–1

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