Page 306 - Advanced engineering mathematics

286 CHAPTER 9 Eigenvalues, Diagonalization, and Special Matrices

THEOREM 9.9

An n × n real symmetric matrix with distinct eigenvalues can be diagonalized by an orthogonal
matrix.

EXAMPLE 9.13
Let
⎛ ⎞
3 0 −2
S = ⎝ 0 2 0 ⎠ .
−20 0
This real, symmetric matrix has eigenvalues 2,−1,4, with corresponding eigenvectors
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 1 2
, and .
⎝ 1 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
0 2 −1
The matrix having these eigenvectors as columns will diagonalize S, but is not an orthogonal
matrix because these eigenvectors do not all have length 1. Normalize the second and third eigen-
vectors by dividing them by their lengths, and then use these unit eigenvectors as columns of an
orthogonal matrix Q:
√
⎛ √ ⎞
01/ 5 2/ 5
Q = 1 0 0 ⎠ .
⎝
√ √
02/ 5 −1/ 5
This orthogonal matrix also diagonalizes S.
9.3.2 Unitary Matrices
We will use the following fact. If W is any matrix, then the operations of taking the transpose
and the complex conjugate can be performed in either order:
t
(W ) = (W) .
t
This is verified by a routine calculation.
It is also straightforward to verify that the operations of taking a matrix inverse, and of taking
its complex conjugate, can be performed in either order.
Now let U be an n × n matrix with complex elements.

We say that U is unitary if the inverse is the conjugate of the transpose (which is the same
as the transpose of the conjugate):
t
−1
U = U .
This means that
t
t
(U) U = U(U) = I n .

EXAMPLE 9.14
√
√
i/ 2 1/ 2
U = √ √ .
−i/ 21/ 2
It is routine to check that U is unitary.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:49 THM/NEIL Page-286 27410_09_ch09_p267-294