Page 323 - A Course in Linear Algebra with Applications

9.1: Symmetric and Hermitian Matrices 307

which equals

1 0 \ / c i B \ ( l 0 \ (a BU X \
0 UZJ\0 A J\0 Uj V° UtA&iJ-
1
This shows that

C
U*AU=( ' B ^y

an upper triangular matrix, as required.
If the matrix A is real symmetric, the argument shows
T
that there is a real orthogonal matrix S such that S AS is
diagonal. The point to keep in mind here is that the eigenval-
ues of A are real by 9.1.1, so that A has a real eigenvector.

The crucial theorem on the diagonalization of hermitian
matrices can now be established.

Theorem 9.1.3 (The Spectral Theorem)
Let A be a hermitian matrix. Then there is a unitary matrix U
such that U*AU is diagonal. If A is a real symmetric matrix,
then U may be chosen to be real and orthogonal.
Proof
By 9.1.2 there is a unitary matrix U such that U*AU = T
is upper triangular. Then T* = U*A*U = U*AU = T, so
T is hermitian. But T is upper triangular and T* is lower
triangular, so the only way that T and T* can be equal is if
all the off-diagonal entries of T are zero, that is, T is diagonal.
The case where A is real symmetric is handled by the
same argument.
Corollary 9.1.4
If A is an n x n hermitian matrix, there is an orthonormal
basis of C n which consists entirely of eigenvectors of A. If
in addition A is real, there is an orthonormal basis of R n
consisting of eigenvectors of A.