Page 307 - Advanced engineering mathematics

9.3 Some Special Types of Matrices 287

If U is a unitary matrix with real elements, then U = U and the condition of being unitary
t
becomes U −1 = U . Therefore a real unitary matrix is orthogonal. In this sense unitary matrices
are the extension of orthogonal matrices to allow complex matrix elements.
Since the rows (and columns) of an orthogonal matrix are mutually orthogonal unit vectors,
we would expect a complex analogue of this condition for unitary matrices. If (x 1 ,··· , x n ) and
n
(y 1 ,··· , y n ) are vectors in R , we can write
⎛ ⎞ ⎛ ⎞
x 1 y 1
x 2 y 2
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
X = ⎜ . ⎟ and Y = ⎜ . ⎟
. .
⎝ . ⎠ ⎝ . ⎠
x n y n
t
and obtain the dot product X·Y as the matrix product X Y, which is the 1×1 matrix (or number)
x 1 y 1 + x 2 y 2 + ··· + x n y n . In particular, the square of the length of X is
2
2
2
t
X X = x + x + ··· + x .
n
1
2
To generalize this to the complex case, suppose we have complex n− vectors (z 1 , z 2 ,··· , z n ) and
(w 1 ,w 2 ,··· ,w n ).Let
⎛ ⎞ ⎛ ⎞
z 1 w 1
z 2 w 2
⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
Z = ⎜ . ⎟ and W = ⎜ . ⎟
. .
⎝ . ⎠ ⎝ . ⎠
z n w n
and define the dot product Z · W by
t
Z · W = Z W.
Then
Z · W = z 1 w 1 + z 2 w 2 + ··· + z n w n .
In this way,
n

2
Z · Z = z 1 z 1 + z 2 z 2 + ··· + z n z n = |z j | ,
j=1
a real number, consistent with the interpretation of the dot product of a vector with itself as
the square of the length. With this as background, we now define the complex analogue of an
n
orthonormal set of vectors in R . We will say that complex n− vectors F 1 ,··· ,F r form a unitary
system if F j · F k = 0if j = k, and each F j has length 1 (that is, F j · F j = 1).
A unitary system is an orthonormal set of vectors when each of the vectors has real
components. With this background, we can state the unitary version of Theorem 9.8.

THEOREM 9.10

A complex matrix U is unitary if and only its row (column) vectors form a unitary system.

We claim that the eigenvalues of a unitary matrix must have magnitude 1.

THEOREM 9.11
Let λ be an eigenvalue of a unitary matrix U. Then |λ|= 1.

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October 14, 2010 14:49 THM/NEIL Page-287 27410_09_ch09_p267-294