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.