Let A be a n × n real symmetric
matrix . Then there exist real numbers
and
real matrices
such that
the following holds :
(i) Each
is an eigenvalue of A.
(ii) Each
is an orthogonal projection matrix, of rank 1, i.e.,
is real symmetric,
=
P and rank () = 1.
By theorem 10.3.4, A has n-real eigenvalues,
,
and an orthonormal basis ,
of
, where
.
Further, if
then
Define
Then, each
is a real symmetric matrix with
That
follows
from the fact that P is orthogonal.
Clearly, rank (
) = 1. Finally,
such that 
is an eigenvalue of A.
is an orthogonal projection matrix, of rank 1, i.e.,
=
P and rank (
,
of
, where
.
Further, if 
follows
from the fact that P is orthogonal.
