Let (S, C) be a matrix pair of order n. The determinant
#math42#
det(SoX + C)
is a polynomial in the indeterminates of X of degree at
most n over the real field. We call this polynomial the
<#175#>indicator polynomial<#175#> of the matrix pair (S, C)
because of the following proposition.
#theorem176#
#proof180#
For SNS-matrix pairs (S, C) with C = O the indicator
polynomial is a homogeneous polynomial of degree n. In
this case Theorem #th:prop#195> is a standard fact about
SNS-matrices.
#lemma196#
#theorem210#
We note for later use that each submatrix of Tn of
order n - 1 has all 1s on its main diagonal.
We now obtain a bound on the number of nonzero entries of
S in a SNS-matrix pair (S, C) in terms of the degree of
the indicator polynomial. We denote the strictly upper
triangular (0,1)-matrix of order m with all 1s above the
main diagonal by Um. The all 1s matrix of size m by
p is denoted by Jm, p.
#proposition227#
#lemma239#
<#251#>Proof<#251#>. Applying integration by parts, we obtain
#math43#
| #tex2html_wrap_indisplay1350#e-2s0t[v2(t) - v2(0)]dt |
= |
#tex2html_wrap_indisplay1353##tex2html_wrap_indisplay1354# - #tex2html_wrap_indisplay1355#e-2s0tv2(t)#tex2html_wrap_indisplay1356# + #tex2html_wrap_indisplay1357##tex2html_wrap_indisplay1358#e-2s0tv(1)(t)v(t)dt |
|
| |
≤ |
#tex2html_wrap_indisplay1361##tex2html_wrap_indisplay1362#e-2s0tv(1)(t)v(t)dt ≤ 0. |
|
Thus
#math44#
#tex2html_wrap_indisplay1364#e-2s0tv2(t)dt ≤v2(0)#tex2html_wrap_indisplay1365# e-2s0tdt = #tex2html_wrap_indisplay1366#v2(0).#tex2html_wrap_indisplay1367##tex2html_wrap_indisplay1368#
#corollary276#
<#1370#>Definition <#1370#>
<#1372#><#465#>Let S be an isolated invariant set with isolating neighborhood N.
An <#283#>index pair<#283#> for S is a pair of compact sets #math45#(N1, N0)
with #math46#N0⊂N1⊂N such that:
#romannum288#
<#465#><#1372#>