Page 325 - A Course in Linear Algebra with Applications

9.1: Symmetric and Hermitian Matrices 309

2
These are orthogonal; to get an orthonormal basis of R , re-
place them by the unit eigenvectors

^OO^T^i

Finally let

which is an orthogonal matrix. The theory predicts that

as is easily verified by matrix multiplication.

Example 9.1.2
Find a unitary matrix which diagonalizes the hermitian matrix

/ 3/2 i/2 0'
A= -i/2 3/2 0
V 0 0 1

where i = ^/—T.

The eigenvalues are found to be 1, 2, 1, with associated
unit eigenvectors

-if y/2\ ( 1/V2 \ / 0 '
l/x/2 , -i/y/2 , 0

Therefore
/ l 0 0'
U*AU = 0 2 0