### Marcin Skrzynski

Institute of Mathematics, Cracow University of Technology, ul. Warszawska 24, 31-155 Krakow, Poland
e-mail: pfskrzyn@cyf-kr.edu.pl

Abstract.   We study linear subspaces L Mn (over an algebraically closed field F of characteristic zero) and their singular sets S(L) defined by S(L) = {A Mn : χ(A + L) is not dense in Fn}, where χ : MnFn is the characteristic map. We give a complete characterization of the subspaces L M2 such that S(L) ≠ M2. We also provide a complete characterization of the singular sets S(L) in the case of n = 2. Finally, we give a characterization of the n-dimensional subspaces L Mn such that S(L) = by means of their intersections with conjugacy classes.

2000 Mathematics Subject Classification.   15A18, 14A10, 14L35.

Key words and phrases.   Characteristic polynomial of a matrix, characteristic map, dominant map, linear space of matrices, triangularizable set of matrices, conjugacy class of a matrix.

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DOI: 10.3336/gm.42.2.04

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