Kyrchei

Student of the Faculty of Mathematics and Informatics of Tavrida National V.I. Vernadsky University (Simferopol) (1984-1988),
Student of the Faculty of Mechanics and Mathematics, Ivan Franko National University of L’viv (1988-1992),
Teacher of Mathematics of Verhne Synevydne secondary school Lviv region (1990-1994),
Postgraduate of Pidstryhach Institute for Applied Problems, of Mechanics and Mathematics (IAPMM) of NAS of Ukraine (1993-1996),
Mathematician of I-th category of IAPMM (2001-2004),
Leading mathematician of IAPMM (2004-2008),
Junior Resercher of IAPMM (2008-2009),
Researcher of IAPMM (2010-2011),
Senior Researcher of IAPMM from 2011 to the present time,
In 2014, he was awarded the title of Senior Research Fellow (Algebra and The Theory of Numbers) from Ministry of Education and Science of Ukraine, equivalent to Associate Professor.

New matrix functionals - row, column immanants(permanents, determinants) of square matrices have been introduced and their properties have been considered. A key theorem about an immanant of a Hermitian matrix has been proved.[ 4]. By using row-column determinants previously introduced in [ 4], properties of the determinant of a Hermitian matrix have been investigated, and determinantal representations of the inverse of a Hermitian coquaternionic matrix have been obtained. With their help, Cramer's rules for left and right systems of linear equations with Hermitian coquaternionic coefficient matrices are obtained. Cramer's rule for a two-sided coquaternionic matrix equation AXB= D (with Hermitian A, B) is given as well. [ 1].

A theory of new matrix functionals: row, column and double determinants of square matrices (noncommutative determinants) has been introduced and developed in the papers [ 14, 20, 21] and the chapter [4]. Relationship between the introduced noncommutative determinants, and other well-known noncommutative determinants (Moore, Study, Dieudonne, Chen) and quasideterminants of Gelfand-Retakh has been obtained [3].
(The column and row determinants can be calculated by Maple. Required packages and examples can be found here).

Determinantal representations of the inverse matrix by analogs of classical adjoint [ 18, 19],, the Moore-Penrose inverse [10, 12], the Drazin inverse [6] , and the W-weighted Drazin inverse [5] have been obtained.
Analogs of Cramer's rule for the solutions ([14, 15]) the minimum norm least squares solutions ([10]), and Drazin inverse solutions ([6]) of right and left systems of linear equations were obtained.
Determinantal representations (analogs of Cramer's rule) of the solutions ([11]), the minimum norm least squares solutions ([8]), the Drazin inverse solutions ([6]), and the W-weighted Drazin inverse solutions ([3]) of some matrix equations have been obtained as well.
Determinantal representations of solutions of the differential quaternion-matrix equations, X' + AX = B and X' + XA = B, where the matrix A is noninvertible, have been considered in ([1, 3]).
The above Identified problems summarized and discussed more completely in the chapters ([1, 3]).

Determinantal representations of the Moore-Penrose inverse [13,17], and the Drazin inverse [7, 16].
Explicit representation formulas (analogs of Cramer's rule) for the minimum norm least squares solutions for systems of linear equations ([13]) and for some matrix equations ([9]), and the Drazin inverse solutions for systems of linear equations ([16]), for some matrix and differential matrix equations ([7]) have been obtained as well..
The above identified problems for generalized inverses of Moore-Penrose, Drazin, and their weighted summarized and discussed more completely in the chapter ([2]).

Ivan Kyrchei