• • • <lk ≤ n. Note that
i ≤ li , for all i ≤ k.
Therefore
max(i, j) ≤ max(li, lj), for i, j ≤ k.
If equality does not hold for some i and j, then by (1) we have,
aij ≥ ali,lj.
If equality does hold for some i and j, then let lt = max(li, lj ). Note that
i < j ⇔ li < lj . So we have t = max(i, j) and therefore lt = t. Hence
lt-1 = t - 1, . . . , l1 = 1. In this case
aij = ali,lj.
Either way, aij ≥ ali ,lj holds. Therefore
m(A(l1, . . . lk)) ≤ m(A(1, . . . k))
= m(A[k])
= δk(A),
as A(l1 , . . . lk) was arbitrary. Hence the result.
Example 14. Consider the matrix
9 91 |
8 |
4 |
3 ʌ | ||
A = |
8 |
6 |
5 |
4 | |
5 |
4 |
4 |
3 | ||
3 |
2 |
3 |
1 |
/ |
The indicated lines help to check that A is pyramidal. Hence by Theorem 13
we find:
δ1(A) = m(A[1]) = 9
δ2 (A) = m(A[2]) = 16
δ3 (A) = m(A[3]) = 20
δ4(A) = m(A[4]) = m(A) = 22.
12