Roksana Słowik

Bijective maps of infinite triangular and unitriangular matrices preserving commutators

We investigate bijective maps of unitriangular matrices which preserve commutators. We show that every such map is a composition of at most four standard maps preserving commutators, of which three are automorphisms. Next, using this result, we obtain an analogous description for maps of triangular matrices preserving commutators. We prove that they are compositions of at most three standard maps, of which two are automorphisms.

Sums of square-zero infinite matrices

We investigate column finite matrices over fields of characteristic different from 2. We prove that every such matrix can be written as a sum of at most 10 square-zero matrices.

Maps on infinite triangular matrices preserving idempotents

We consider maps of triangular infinite matrices over fields of characteristic different from 2, that have the following property: is idempotent if and only if is idempotent for all . We show that every such map is a sum of compositions of some standard maps.

Corrigendum to ‘Every infinite triangular matrix is similar to a generalized infinite Jordan matrix’ (Linear Multilinear Algebra 65 (2017), 1362–1373)

The author regrets that the printed version of the above article contains some errors in the proof of Proposition 2.3 and would like to apologize for that. Moreover, to fill this gap at least a little bit, the author would like to formulate and briefly sketch the proof of the following. Lemma 0.7: Suppose that F is any field and that a ∈ NT∞(F). Let m1, m2, m3, . . . denote the indices of nonzero columns of a ordered increasingly. If for all i ∈ N the following condition holds max 1≤j<mi+1 { ajmi+1 = 0 } = mi, then there exists t ∈ T∞ (F) such that t−1at is equal to a generalized J∞(0).

Every infinite triangular matrix is similar to a generalized infinite Jordan matrix

In this paper, we introduce the notion of a generalized infinite Jordan matrix which is an infinite analogon of a finite Jordan matrix. The main result states that for every upper triangular infinite matrix a, there exists a column finite matrix x such that is a generalized infinite Jordan matrix.

Involutions in triangular groups

We describe involutions, i.e. elements of order 2, in the groups T n (K) – of upper triangular matrices of dimension n (n ∈ ℕ), and T ∞(K) – of upper triangular infinite matrices, where K is a field of characteristic different from 2. Using the obtained result, we give a formula for the number of all involutions in T n (K) in the case when K is a finite field.

Corrigendum to ‘Sums of square-zero infinite matrices’ (Linear Multilinear Algebra 64 (2016), 1760–1768)

Clearly, t = v1 + v2, where v1 = t + u, v2 = −u. Since (v1)i,i+1 = 0, there exists such invertible triangular matrix u that that u−1v1u is equal to J∞ = ∑∞ i=1 ei,i+1, the latter is equal to the sum ṽ1 + ṽ2, ṽ1 = ∑∞ i=1 e2i−1,2i, ṽ2 = ∑∞ i=1 e2i,2i+1, i.e. the sum of two square-zero matrices. The matrix v2 is a negative direct sum of the matrices of the form J∗ = ∑∗ i=1 ei,i+1, where ∗ is either a natural number or it is ∞. Each of the matrices J∗ can be written as a sum of two square-zero matrices, so v2 as a direct sum of them also can be written in such way.

Epimorphisms of the ring of infinite triangular matrices

We consider the ring of all upper triangular matrices over any field of at least three elements. We give a description of all epimorphisms of this ring. We show that they are of almost the same form as group epimorphisms of its multiplicative group.

How to construct a triangular matrix of a given order

In this article we consider the elements in upper triangular finite and infinite dimensional matrix groups over fields, whose order is equal to (). For the case when the characteristic of does not divide, we give a description of such elements. Next using this criterion, we show how to find the number of these elements in finite dimensional groups when is a finite field.

Maps on upper triangular matrices preserving zero products

Consider Tn(F)—the ring of all n × n upper triangular matrices defined over some field F. A map φ is called a zero product preserver on Tn(F) in both directions if for all x, y ∈ Tn(F) the condition xy = 0 is satisfied if and only if φ(x)φ(y) = 0. In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a composition of three types of maps. The first of them is a conjugacy, the second one is an automorphism induced by some field automorphism, and the third one transforms every matrix x into a matrix x′ such that {y ∈ Tn(F): xy = 0} = {y ∈ Tn(F): x′y = 0}, {y ∈ Tn(F): yx = 0} = {y ∈ Tn(F): yx′ = 0}.