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Triangular ranks do not bound minimum ranks in matrix completion

Yaroslav Shitov

Vol. 6 (2024), No. 3, 543–548
Abstract

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 rank mr (A) is the smallest value of the ranks of all matrices obtained from A by replacing the symbols with arbitrary elements in 𝔽. A triangular relaxation T is obtained by placing at several specified entries of A so that

(a b )

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.

Keywords
matrix completion, triangular minimal rank
Mathematical Subject Classification
Primary: 15A03, 15A83
Milestones
Received: 26 July 2023
Revised: 28 April 2024
Accepted: 18 June 2024
Published: 30 September 2024
Authors
Yaroslav Shitov
Moscow
Russia