Grigore Calugareanu: Nil-clean integral $2\times 2$ matrices: the elliptic case, 239-250

Abstract:

Nil-clean integral $2\times 2$ matrices are investigated starting with a characterization which involves a Diophantine second degree equation. Mainly similarity classes of matrices A with $1-4\det (A)<0$ are determined (that is, the elliptic case). The strongly nil-clean matrices are completely determined and a large class of uniquely nil-clean matrices is found. In particular, a class of uniquely nil-clean matrices which are not strongly nil-clean is found.

Key Words: Uniquely nil-clean, strongly nil-clean, clean, quadratic Diophantine equation.

2010 Mathematics Subject Classification: Primary 15B36, 16U99, 11D09, 11D45.