Abstract

Let $\mathbb{𝔽}$ be a field, and let $A$ be a matrix with entries in $\mathbb{𝔽}\cup \{\ast \}$, where $\ast$ is a placeholder symbol for nonspecified entries. The minimum rank $\mathrm{mr}<!--FUNCTION\; APPLICATION-->(A)$ is the smallest value of the ranks of all matrices obtained from $A$ by replacing the $\ast$ symbols with arbitrary elements in $\mathbb{𝔽}$. A triangular relaxation $T$ is obtained by placing $\ast$ at several specified entries of $A$ so that

does not appear as a submatrix of $T$ with any $a,b$ in $\mathbb{𝔽}$. We show that $\mathrm{mr}<!--FUNCTION\; APPLICATION-->(A)$ can be arbitrarily large even if $\mathrm{mr}<!--FUNCTION\; APPLICATION-->(T)\le 1$ for any triangular relaxation $T$ of $A$. This answers a question asked by Johnson and Whitney in 1991.

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