Page 830 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

EXAMPLE 5 Unitary Diagonalization
The matrix

is unitarily diagonalizable because it is Hermitian and therefore normal. Find a matrix that unitarily diagonalizes .

Solution

The characteristic polynomial of A is

so the characteristic equation is

and the eigenvalues are and .
By definition,

will be an eigenvector of A corresponding to if and only if is a nontrivial solution of

(4)

To find the eigenvectors corresponding to , we substitute this value in 4:

Solving this system by Gauss–Jordan elimination yields (verify)

Thus the eigenvectors of A corresponding to are the nonzero vectors in of the form

Thus this eigenspace is one-dimensional with basis (5)
In this case the Gram–Schmidt process involves only one step: normalizing this vector. Since
the vector