A note on preserving the spark of a matrix

Marcin Skrzyński

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).

Keywords


spark of a matrix, linear preserver

Mathematics Subject Classification (2010)


primary 15A03, secondary 15A86

References


Beasley, L.B. and T.J. Laffey. "Linear operators on matrices: the invariance of rank-k matrices." Linear Algebra Appl. 133 (1990): 175-184.

Pierce, S., et al. "A survey of linear preserver problems." Linear and Multilinear Algebra 33.1-2 (1992): 1-129.

Donoho, D.L. and M. Elad. "Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization." Proc. Natl. Acad. Sci. USA 100.5 (2003): 2197-2202.


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