Page 57 - Textos de Matemática Vol. 39

TRIANGLE SIZE AND RANK FOR ZERO-NONZERO PATTERNS 47
We have also verified, via exhaustion of cases (based upon ideas of [2]) that t(5, 5, 4) = 4. This allows us to present the following table, in which Y indicates that t(m,n,r) = r, and N indicates that t(m,n,r) < r. The table covers the cases m = n, but many cases m < n may be deduced from these. The two ?’s indicate that the cases t(6, 6, 4) and t(6, 6, 5) are currently unresolved.
m=n
1 −→ 2 −→ 3 −→ 4 −→
r 5 ↘ −→ 6 ↘ −→ 7 ↘ −→ 8 ↘ −→ 9 ↘ −→
. .
We close by noting an example of interest, relative to the upper bounds on minimum rank, given in [2] and based upon design of null spaces.
Example 1.12. For the 7-by-7 pattern
⎡⎢ ∗ ∗ 0 0 0 0 0 ⎤⎥ ⎢ ∗ ∗ ∗ 0 0 0 0 ⎥ ⎢ 0 ∗ ∗ ∗ 0 ∗ 0 ⎥
P=⎢0 0 ∗ ∗ ∗ 0 0⎥, ⎢000∗∗00⎥ ⎣00∗00∗∗⎦
00000∗∗
from [2], we have the bounds mr(P)≤7+1−r1=7+1−c1=7+1−2=6, the best available. However, as each of P[{1,2},{1,2}], P[{2,3,4,6},{3}], P[{3},{2,4,6}], P[{4,5},{4,5}], and P[{6,7},{6,7}] is a full matrix, there are matrices of pattern P that are a sum of 5 rank 1 matrices, such as
⎡⎢ 1 1 0 0 0 0 0 ⎤⎥ ⎢ 1 1 1 0 0 0 0 ⎥ ⎢ 0 1 1 1 0 1 0 ⎥
A=⎢0 0 1 1 1 0 0⎥. ⎢0001100⎥ ⎣0010011⎦
0000011
1
2
Y
3
4
5
Y
6
7
8
9
10
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
?
Y
Y
Y
Y
Y
Y
?
N
N
N
N
Y
N
N
N
Y
N
N
N
Y
N
N
Y
N
Y