Completion of a partial integral matrix to a unimodular matrix. (English) Zbl 1090.15013
The paper deals with unimodular matrix completion problem, that is, when a partial integral matrix has a unimodular matrix completion.
Given an \(n \times n\) integral unimodular matrix \(A\), the author characterizes submatrices of \(A\) via the number of invariant factors equal to 1.
By using the above characterization, he proves that if \(n\) entries of an \(n \times n\) partial integral matrix are prescribed and these entries do not constitute a row or a column, then this matrix can be completed to a unimodular matrix. As a trivial consequence, every \(n \times n\) partial integral matrix with \(n-1\) prescribed entries has a unimodular matrix completion.
Given an \(n \times n\) integral unimodular matrix \(A\), the author characterizes submatrices of \(A\) via the number of invariant factors equal to 1.
By using the above characterization, he proves that if \(n\) entries of an \(n \times n\) partial integral matrix are prescribed and these entries do not constitute a row or a column, then this matrix can be completed to a unimodular matrix. As a trivial consequence, every \(n \times n\) partial integral matrix with \(n-1\) prescribed entries has a unimodular matrix completion.
Reviewer: Juan Ramon Torregrosa Sanchez (Valencia)
MSC:
| 15A29 | Inverse problems in linear algebra |
| 15B36 | Matrices of integers |
| 11C20 | Matrices, determinants in number theory |
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