On rank-critical matrix spaces

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Journal Article
Differential Geometry and its Application, 2017, 55 pp. 68 - 77
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© 2017 Elsevier B.V. A matrix space of size m×n is a linear subspace of the linear space of m×n matrices over a field F. The rank of a matrix space is defined as the maximal rank over matrices in this space. A matrix space A is called rank-critical, if any matrix space which properly contains it has rank strictly greater than that of A. In this note, we first exhibit a necessary and sufficient condition for a matrix space A to be rank-critical, when F is large enough. This immediately implies the sufficient condition for a matrix space to be rank-critical by Draisma (2006) [5], albeit requiring the field to be slightly larger. We then study rank-critical spaces in the context of compression and primitive matrix spaces. We first show that every rank-critical matrix space can be decomposed into a rank-critical compression matrix space and a rank-critical primitive matrix space. We then prove, using our necessary and sufficient condition, that the block-diagonal direct sum of two rank-critical matrix spaces is rank-critical if and only if both matrix spaces are primitive, when the field is large enough.
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