Page 356 - A Course in Linear Algebra with Applications

340 Chapter Nine: Advanced Topics

have some common properties, as similar matrices do. How-
ever, whereas similar matrices have the same eigenvalues, this
is not true of congruent matrices. For example, the matrix

( o - a )

has eigenvalues 2 and —3, but the congruent matrix

(i!)G-°)(i 9-('-?)

has eigenvalues —2 and 3.
Notice that, although the eigenvalues of these congruent
matrices are different, the numbers of positive and negative
eigenvalues are the same for each matrix. This is an instance
of a general result.

Theorem 9.3.5 (Sylvester's Law of Inertia)
Let A be a real symmetric n x n matrix and S an invertible
T
n x n matrix. Then A and S AS have the same numbers of
positive, negative and zero eigenvalues.

Proof
Assume first of all that A is invertible; this is the essential
case. Recall that by 7.3.6 it is possible to write S in the form
QR where Q is real orthogonal and R is real upper triangular
with positive diagonal entries; this was a consequence of the
Gram-Schmidt process.
The idea of the proof is to obtain a continuous chain
of matrices leading from S to the orthogonal matrix Q; the
T
l
point of this is that Q AQ — Q~ AQ certainly has the same
eigenvalues as A. Define
S{t)=tQ + {l-t)S,

where 0 < t < 1. Thus 5(0) = S while 5(1) = Q. Now write
U = tl + (1 - t)R, so that S{t) = QU. Next U is an upper