Graphics Reference
In-Depth Information
Step 2.
To make A
1
congruent to a matrix A
2
, which has a
11
nonzero.
Let F = (f
ij
) be the elementary matrix defined by
f
st
=
1
,
if
s
=
t
,
s
π
1
or i
,
=
1
,
if
s
=
1
,
t
=
i
=
1
,
if
s
=
i
,
t
=
1
=
0
,
otherwise
.
Then A
2
= FA
1
F
T
is the matrix obtained from A
1
by interchanging the first and ith
diagonal element.
Step 3.
To make A
2
congruent to a matrix A
3
in which the only nonzero element
in the first row or first column is a
11
.
Step 3 is accomplished via elementary matrices like in Step 1 that successively
add multiples of the first row to all the other rows from 2 to n and the same multi-
ples of the first column to the other columns.
After Step 3, the matrix A
3
will have the form
a
0
Ê
Ë
ˆ
¯
11
0
B
where B is a symmetric (n - 1) ¥ (n - 1) matrix. Repeating Steps 1-3 on the matrix
B and so on will show that A is congruent to a diagonal matrix with the first r diag-
onal entries nonzero. By interchanging the diagonal entries like in Step 2 if necessary,
we may assume that all the positive entries come first. This shows that A is congru-
ent to a diagonal matrix
(
)
GDd
=
,...,
d d
,
-
,...,
-
d
, ,..,
00
,
1
s
s
+
1
r
where d
i
> 0. If
1
1
Ê
Ë
ˆ
¯
HD
d
=
,...,
, ,...,
00
,
d
r
1
then HGH
T
has the desired form. To see why s is uniquely determined see [Fink72].
One nice property of the proof of Theorem 1.9.11 is that it is constructive.
1.9.12. Example.
To show that the matrix
Ê
ˆ
10 0
Á
Á
Á
Á
Á
˜
˜
˜
˜
˜
1
2
A =
00
-
1
2
0
-
0
Ë
¯
is congruent to a diagonal one with ±1s or 0 on the diagonal.
