Sylvester's criterion
Exercise 1. Suppose where
and
. Then
is positive if and only if all
are positive.
Proof. For any we have
Let
Then
This is positive for all
if and only if
Exercise 2 (modified Gaussian elimination). Suppose that is a real symmetric matrix with nonzero leading principal minors
. Then
where
and
.
Proof. Review the transformation applied in Exercise 1 to obtain a triangular form. In that exercise, we eliminated element below
by premultiplying
by the matrix
Now after this we can post-multiply
by the matrix
Because of the assumed symmetry of
we have
so this will eliminate element
to the right of
, see Exercise 2. Since in the first column
is already
, the diagonal element
will not change.
We can modify Exercise 1 by eliminating immediately after eliminating
The right sequencing of transformations is necessary to be able to apply Exercise 1: the matrix used for post-multiplication should be the transpose of the matrix used for premultiplication. If
then
which means that premultiplication by
should be followed by post-multiplication by
In this way we can make zero all off-diagonal elements. The resulting matrix
is related to
through
Theorem (Sylvester) Suppose that is a real symmetric matrix. Then
is positive if and only if all its leading principal minors are positive.
Proof. Let's assume that all leading principal minors are positive. By Exercise 2, we have where
It remains to apply Exercise 1 above to see that
is positive.
Now suppose that is positive, that is
for any
Consider cut-off matrices
The corresponding cut-off quadratic forms
are positive for nonzero
It follows that
are non-singular because if
then
Hence their determinants
are nonzero . This allows us to apply the modified Gaussian elimination (Exercise 2) and then Exercise 1 with
By Exercise 1 consecutively
Exercise 3. is negative if and only if the leading principal minors change signs, starting with minus:
Proof. By definition, is negative if
is positive. Because of homogeneity of determinants, when we pass from
to
the minor of order
gets multiplied by
Thus, by Sylvester's criterion
is negative if and only if
as required.
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