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Séminaire Lotharingien de Combinatoire, B52f (2004), 11 pp.

# Marc Fortin and Christophe Reutenauer

# Commutative/Noncommutative Rank of Linear Matrices and Subspaces of
Matrices of Low Rank

**Abstract.**
A space of matrix of low rank is a vector space of rectangular matrices
whose maximum rank is stricly smaller than the number of rows and the
numbers of columns. Among these are the compression spaces, where the
rank condition is garanteed by a rectangular hole of 0's of appropriate
size. Spaces of matrices are naturally encoded by linear matrices. The
latter have a double existence: over the rational function field, and
over the free field (noncommutative). We show that a linear matrix
corresponds to a compression space if and only if its rank over both
fields is equal. We give a simple linear-algebraic algorithm in order
to decide if a given space of matrices is a compression space. We give
inequalities relating the commutative rank and the noncommutative rank
of a linear matrix.

Received: May 21, 2004.
Accepted: November 8, 2004.
Final Version: November 16, 2004.

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