Let A be n × n real symmetric
matrix such that all its eigenvales are distinct. Then, there
exists an orthogonal matrix P such that , where D is a diagonal matrix with diagonal entries being the
eigenvalues of A.

is invertible and
,
is diagonal with diagonal entries ..........
Further, by theorem 10.3.1,
is
an orthogonal set. Hence P is in fact an orthogonal
matrix.