Glasnik Matematicki, Vol. 42, No.2 (2007), 291-299.

ON LINEAR SUBSPACES OF M_n AND THEIR SINGULAR SETS RELATED TO THE CHARACTERISTIC MAP

Marcin Skrzynski

Abstract.   We study linear subspaces L M_n (over an algebraically closed field F of characteristic zero) and their singular sets S(L) defined by S(L) = {A M_n : χ(A + L) is not dense in Fⁿ}, where χ : M_n → Fⁿ 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 M_n 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.

DOI: 10.3336/gm.42.2.04

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