Let
F be a field, and
let
A be a matrix
with entries in
F∪{∗},
where
∗
is a placeholder symbol for nonspecified entries. The
minimum rankmr(A)
is the smallest value of the ranks of all matrices obtained from
A by replacing the
∗ symbols with arbitrary
elements in
F. A
triangularrelaxation T is obtained
by placing
∗ at several
specified entries of
A
so that
(∗ab∗)
does not appear as a submatrix of
T
with any
a,b in
F. We show that
mr(A) can be arbitrarily
large even if
mr(T)≤1 for any
triangular relaxation
T
of
A.
This answers a question asked by Johnson and Whitney in 1991.
