In this paper we characterize all $2times 2$ idempotent and nilpotent matrices over an integral domain and then we characterize all $2times 2$ strongly nil-clean matrices over a PID. Also, we determine when a $2times 2$ matrix  over a UFD is nil-clean.

The problem formulated in the title is investigated. The case of nilpotent matrices of size at most 4 allows a unitary treatment. The numerical range of a nilpotent matrix M of size at most 4 is circular if and only if the traces trM M and trM M are null. The situation becomes more complicated as soon as the size is 5. The conditions under which a 5 5 nilpotent matrix has circular numerical ran...

A zero-nonzero pattern A is said to be potentially nilpotent over a field F if there exists a nilpotent matrix with entries in F having zero-nonzero pattern A. We explore the construction of potentially nilpotent patterns over a field. We present classes of patterns which are potentially nilpotent over a field F if and only if the field F contains certain roots of unity. We then introduce some ...

Let B be a nilpotent matrix and suppose that its Jordan canonical form is determined by a partition λ. Then it is known that its nilpotent commutator NB is an irreducible variety and that there is a unique partition μ such that the intersection of the orbit of nilpotent matrices corresponding to μ with NB is dense in NB. We prove that map D given by D(λ) = μ is an idempotent map. This answers a...

The paper studies algebraic shift equivalence of matrices over n-variable polynomial rings over a principal ideal domain D(n ≤ 2). It is proved that in the case n = 1, every non-nilpotent matrix over D[x] is algebraically strong shift equivalent to a nonsingular matrix. In the case n = 2, an example of non-nilpotent matrix over R[x, y, z] = R[x][y, z], which can not be algebraically shift equiv...

There are some new developments on Plancherel formula and growth of matrix coefficients for unitary representations of nilpotent Lie groups. These have several consequences for the geometry of weakly symmetric spaces and analysis on parabolic subgroups of real semisimple Lie groups, and to (infinite dimensional) locally nilpotent Lie groups. Many of these consequences are still under developmen...

Abstract. In this paper we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GLn(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log2 n⌉ square-zero matrices and also find the necessary number of square-zer...

A sign pattern is a matrix with entries in {+,−, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentially...

Reopening a cold case, Inspector Echelon, high-ranking in the Row Operations Center, is searching for lost linear map, known to be nilpotent. When partially decomposed matrix unearthed, he reconstructs its reduced form, finding it singular. But were origins nilpotent?