A note on preserving the spark of a matrix
Abstract
Let Mm×n(F) be the vector space of all m×n matrices over a field F. In the case where m ≥n, char(F) ̸= 2 and F has at least five elements, we give a complete characterization of linear maps Φ: Mm×n(F) →Mm×n(F) such that spark(Φ(A)) = spark(A) for any A ∈Mm×n(F).
References
L.B. Beasley, T.J. Laffey, Linear operators on matrices: the invariance of rank-k matrices, Linear Algebra Appl. 133 (1990), 175–184.
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L.B. Beasley (ed.), C.-K. Li (ed.), M.H. Lim (ed.), R. Loewy (ed.), B.R. McDonald (ed.), S. Pierce (ed.), A survey of linear preserver problems, Linear Multilinear Algebra 33 (1992) no. 1-2, Gordon and Breach Science Publishers, Yverdon, 1992, pp. 1–129.
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D.L. Donoho, M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization, Proc. Natl. Acad. Sci. USA 100 (2003) no. 5, 2197–2202. Cited on 63. Institute of Mathematics Cracow University of Technology ul. Warszawska 24 31-155 Kraków Poland E-mail: pfskrzyn@cyf-kr.edu.pl Received: October 12, 2014; final version: April 26, 2015; available online: May 6, 2015.
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