next up previous
Next: References Up: Sufficient Conditions for Multiply Previous: Non-positive eigenvalues

Negative eigenvalues

The following lemma will be needed in the proof of Theorem 3.1, at a point where we will apply Lemma 2.3.

  lemma138

Proof: Let tex2html_wrap_inline928 , tex2html_wrap_inline930 be vectors such that tex2html_wrap_inline932 if and only if tex2html_wrap_inline934 , i=1, tex2html_wrap_inline438 , n, and tex2html_wrap_inline942 if and only if tex2html_wrap_inline944 , j=1, tex2html_wrap_inline438 , m. Then a vector tex2html_wrap_inline942 is in tex2html_wrap_inline954 if and only if there are constants tex2html_wrap_inline956 so that tex2html_wrap_inline958 . For this we need

displaymath900

so that d is in tex2html_wrap_inline954 if and only if the vector

displaymath901

is in the span of the vectors

displaymath902

This characterization of the image of C under the projection tex2html_wrap_inline966 and the fact that inner product is continuous is enough to yield the result.

We now develop a criterion slightly different from that of the previous section. We suppose that tex2html_wrap_inline968 consists of k negative eigenvalues (counting multiplicity), that ker(A) is finite dimensional (this will be implied by the requirement that tex2html_wrap_inline974 is trivial), and tex2html_wrap892 for some tex2html_wrap_inline712 . Suppose that tex2html_wrap_inline978 is spanned by tex2html_wrap_inline980 . We may assume that tex2html_wrap_inline982 , tex2html_wrap_inline438 , tex2html_wrap_inline986 are orthogonal to ker(A) by projecting onto tex2html_wrap_inline532 . Define E to be

displaymath903

  theorem175

Proof: Again we need to show that A is strongly positive on tex2html_wrap_inline706 . By orthogonalizing the matrix E as in Lemma 2.2, we may assume that tex2html_wrap_inline1024 if tex2html_wrap_inline1026 and that tex2html_wrap_inline1028 . Take an tex2html_wrap_inline716 . Then x=y+z, where tex2html_wrap_inline1034 and tex2html_wrap_inline1036 . One easily sees that tex2html_wrap_inline1038 .

By arguing as in Theorem 2.1, one can show that there are constants tex2html_wrap_inline724 , tex2html_wrap_inline438 , tex2html_wrap_inline728 so that tex2html_wrap893 , where v is orthogonal to tex2html_wrap_inline732 , tex2html_wrap_inline438 , tex2html_wrap_inline736 . In addition, v is orthogonal to ker(A), since y and the tex2html_wrap_inline1060 's are. Substituting,

displaymath904

Since tex2html_wrap_inline1062 , we have tex2html_wrap894 , so that tex2html_wrap895 holds for all j. Therefore

displaymath905

with the last inequality following from the fact that v is orthogonal to all non-positive eigenvectors of A. We must now relate the length of v to that of x. As before, we use Lemma 2.3. Since tex2html_wrap_inline974 is trivial, there is a tex2html_wrap_inline1076 so that tex2html_wrap_inline1078 , where tex2html_wrap_inline1080 is independent of x.

For a similar inequality relating tex2html_wrap_inline1084 and tex2html_wrap_inline818 , we must first show that the orthogonal projection of tex2html_wrap_inline706 onto tex2html_wrap_inline532 has only the trivial intersection with the span of tex2html_wrap_inline1092 . Suppose that some tex2html_wrap_inline1094 is also in the orthogonal projection of tex2html_wrap_inline706 onto tex2html_wrap_inline532 . There is a vector tex2html_wrap_inline1100 so that tex2html_wrap_inline1102 . Then for each j from 1 to k,

displaymath906

so that tex2html_wrap_inline1108 is the zero vector. To apply Lemma 2.3 we must also show that the image of tex2html_wrap_inline706 under orthogonal projection onto tex2html_wrap_inline532 is closed. This follows from Lemma 3.1. Therefore, there is some tex2html_wrap_inline1114 (not depending on x or y) so that tex2html_wrap_inline1120 . We now obtain that

displaymath907

for all tex2html_wrap_inline716 , so that A is strongly positive on tex2html_wrap_inline706 . As above, this is sufficient for the result to follow.

note213

Acknowledgments. I would like to thank John Maddocks for suggesting that I extend the results of [6]. I would also like to thank Thomas Schlumprecht for a useful conversation relating to Lemma 3.1.


next up previous
Next: References Up: Sufficient Conditions for Multiply Previous: Non-positive eigenvalues

Tom Vogel
Tue Aug 20 10:23:22 CDT 1996