Page 286 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes We have
(U |U ) = ( |U*U )
= ( |I )
= ( | )

for all , .

n×1
Example 5: Consider C with the inner product (X|Y) = Y*X. Let A be an n × n matrix
over C, and let U be the linear operator defined by U(X) = AX. Then
(UX|UY) = (AX|AY) = Y*A*AX
for all X, Y. Hence, U is unitary if and only if A*A = I.

Definition: A complex n × n matrix A is called unitary, if A*A = I.
Theorem 4: Let V be a finite-dimensional inner product space and let U be a linear operator on V.
Then U is unitary if and only if the matrix of U in some (or every) ordered orthonormal basis is
a unitary matrix.
Proof: At this point, this is not much of a theorem, and we state it largely for emphasis. If  = { ,
1
…, } is an ordered orthonormal basis for V and A is the matrix of U relative to , then A*A =
n
I if and only if U*U = I. The result now follows from Theorem 3.
Let A be an n × n matrix. The statement that A is unitary simply means

(A*A) =
jk jk
n
or A A rk = jk
rj
r 1
In other words, it means that the columns of A form an orthonormal set of column matrices,
with respect to the standard inner product (X|Y) = Y*X. Since A*A = I if and only if AA* = I, we
see that A is unitary exactly when the rows of A comprise an orthonormal set of n-tuples in C
n
(with the standard inner product). So, using standard inner products, A is unitary if and only if
the rows and columns of A are orthonormal sets. One sees here an example of the power of the
theorem which states that a one-sided inverse for a matrix is a two-sided inverse. Applying this
theorem as we did above, say to real matrices, we have the following: Suppose we have a square
array of real numbers such that the sum of the squares of the entries in each row is 1 and distinct
rows are orthogonal. Then the sum of the squares of the entries in each column is 1 and distinct
columns are orthogonal. Write down the proof of this for a 3 × 3 array, without using any
knowledge of matrices, and you should be reasonably impressed.
t
Definition: A real or complex n × n matrix A is said to be orthogonal, if A A = I.
A real orthogonal matrix is unitary; and, a unitary matrix is orthogonal if and only if each of its
entries is real.

Example 6: We give some examples of unitary and orthogonal matrices.

(a) A 1 × 1 matrix [c] is orthogonal if and only if c = ± 1, and unitary if and only if cc = 1. The
i
latter condition means (of course) that |c| = 1, or c = e , where is real.
(b) Let
a b
A = .
c d

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