Education: Ivan Franko
National University,
L’viv, Ukraine
(Speciality: Mathematician, specialization:
Mathematical Analysis, 1981).
Scientific degree: Candidate of Science (Ph.D.), Kiev Taras Shevchenko
State University,
Department of Mechanics and Mathematics.
Dissertation
"Questions of factorization of matrices polynomials over an arbitrary
field". (01.01.06 –
algebra and number theory, 1991).
Individual Soros grant, 1996.
Diploma: Senior Researcher (Algebra and
Number Theory), Kiev Taras Shevchenko
State University, 2000.
Position: Senior
Researcher Worker (April
1997 - present).
In Institute: April 1987 - present.
Scientific profiles
ORCID: https://orcid.org/0000-0001-5539-7904
Scopus: https://www.scopus.com/authid/detail.uri?authorId=16414769100
Google Scholar: https://nbuviap.gov.ua/bpnu/bpnu_profile.php?bpnuid=BUN0020621
Research interests:
Linear algebra, Theory of
matrices, Matrices over function rings.
Direction of scientific researches: factorization
of matrices over function rings; solvability of matrix equations over fields
and commutative rings; investigation a structure of matrices and their
canonical forms over fields and commutative rings with respect to the
similarity and equivalence transformations.
Main scientific results:
1. We present conditions under which a polynomial matrix 
of order 
over a field 
and 
, can be factorized in the form 
, where 
is a monic polynomial matrix and which has a
characteristic polynomial 
. In
the case when the desirbed
divisor 
exists, a method of constructing it is specified. The conditions under which the
matrix is uniquely determined by its characteristic
polynomial 
are presented.
A
necessary and, for certain classes
of polynomial matrices, sufficient
conditions are established
for the existence
of a common
monic
divisor with a prescribed characteristic polynomial of
polynomial matrices and 
.
In the case
when the described
common
divisor
exists,
a method of constructing it is proposed.
2. Let 
be the Smith normal form of a matrix 
, (
) over a a
domain of principal ideals 
. Matrices 
and 
over a domain of principal ideals possess the multiplicative property of Smith
normal forms if the Smith normal form of the product of matrices 
is equal to the product of the Smith normal forms
of matrices and , i.e. 
. Necessary and sufficient
conditions are established when the product is satisfied . We described the structure of the
factorization of a matrix 
, that has the property of multiplicativity 
. The structure was investigated for
a matrix and their divisors over a domain of principal ideals in general case
also.
3. We study the structure of matrices over a domain of principal ideals with respect to the similarity and equivalence
transformations. It is said that a matrix of order is diagonalized, if
it is reduced by the similarity transformation to a diagonal form. We establish necessary and sufficient conditions for the diagonalizability
of matrices over a domain
of principal ideals.
The conditions are determined, under which the matrix is similar to the companion matrix of its
characteristic polynomial. The problems of existence of common eigenvectors and
simultaneous triangularization of a pair of matrices
over a domain of principal ideals with quadratic minimal polynomials are
investigated. The necessary and sufficient conditions of simultaneously triangularization of a pair of matrices with quadratic
minimal polynomials are obtained. As a result, the approach offered provides
the necessary and sufficient conditions of simultaneously triangularization
of pairs of idempotent matrices and pairs of involutory
matrices over a domain of principal ideals.
We give the
canonical form with respect to semiscalar equivalence
for a matrix pencil A(l) = A0 l - A1, where A0 and A1 are n × n matrices over F , and A0
is nonsingular.
4. We established conditions
for solvability of the Riccati
matrix algebraic equation in terms
of the ranks
of matrices constructed in a certain way by
using the coefficients of this equation. We established conditions for the existence
of a unique solution of the matrix polynomial equation A(λ)X(λ) − Y(λ)B(λ) = C(λ) over an
arbitrary field. We propose new necessary and sufficient conditions for the
solvability of a system of linear equations Ax=b
over the domain of principal ideals in terms of (right) Hermite
normal forms of the matrix [ A 0 ]
and [A b ] and an algorithm for the
solution of this system.
Some specialized publications
61. Prokip Volodymyr.
On solutions of matrix equation AX= B over a Bezout
domain // The Electronic Journal of Linear Algebra. - 2025. - V. 41. - P.
181-192. https://doi.org/10.13001/ela.2025.8985 (Scopus, Q2)
60. Prokip, V.M., Mel’nyk,
O.M. and Kolyada, R.V. On the Divisibility with
Remainder of Polynomial Matrices Over an Arbitrary Field // Journal of
Mathematical Sciences. – 2025. – 291, Issue 5. – P. 586–607.
https://doi.org/10.1007/s10958-025-07839-5 (Scopus, Q3)
59. Prokip V.M., Melnyk
O.M., Kolyada R.V. On divisibility with remainder of
polynomial matrices over an arbitrary field // Mat. Metody
Fiz.–Mekh. Polya. – 2023. – 66, ¹ 1–2. – P. 23–39 (in
Ukrainian).
58. Prokip V. M. A note on semiscalar
equivalence of polynomial matrices // Electronic Journal of Linear Algebra. –
2022. – v. 38. – P. 195–203. (Scopus, 0.823, Q2)
57. Prokip V.Ì. Canonical form of involutory matrices over the domain of principal ideals
with respect to similarity transformations. // Journal of Mathematical Sciences
− 2021. − 258, No. 4 − P. 437–445. https://doi.org/10.1007/s10958–021–05558–1
(Scopus, 0.603, Q3)
56. Prokip Volodymyr
M. On the semi–scalar equivalence of polynomial matrices over a field //
Modeling, Control and Information Technologies: Proceedings of International
scientific and practical conference. – 2021. – ¹.5. –
P. 80–82. https://doi.org/10.31713/MCIT.2021.25
55. Prokip V. Equivalence of Polynomial Matrices
over a Field // arXiv preprint arXiv:
2003.05041, 2020.
54. Prokip V. M. Equivalence of Polynomial
Matrices over a Field // Hot Topics in Linear Algebra. Chapter 6. 2020. Ð.
205–232.
53. Prokip V. On solvability of the matrix
equation AXB = C over a principal ideal
domain // Modeling, Control and Information Technologies: Proceedings of
International scientific and practical conference. – 2020. – ¹. 4. – Ñ. 47–50.
52. Prokip
V. Ì. On the divisibility of matrices with remainder over the domain of principal
ideals // Journal of Mathematical Sciences − 2019. − 243, No.1 −
P. 45–55. https://doi.org/10.1007/s10958–019–04524–2 (Scopus, 0.529, Q3)
51. Prokip
V. M. On the Solvability of a System of Matrix Equations AX = B
and BY = A Over Associative
Rings. // Journal of Mathematical Sciences. – 2019. – 238. – P. 22–31.
https://doi.org/10.1007/s10958-019-04215 (Scopus, 0.529, Q3)
50. Prokip
V. Ì. Structure of Rank-One Matrices Over the Domain
of Principal Ideals Relative to Similarity Transformations. // Journal of
Mathematical Sciences. – 2019. – 236. – P. 71–82.
https://doi.org/10.1007/s10958-018-4098-0 (Scopus, 0.529, Q3)
49. Kolyada
R. V., Melnyk O. M., Prokip
V. M. About square roots of matrices over an arbitrary field // Scientific papers UAP. – 2019. – 59, ¹ 2. – P. 56–64.
48. Prokip
V.M. The canonical form of involutory matrices over
the principal ideal domain with respect to similarity transformation // Mat. Metody Fiz.-Mekh.
Polya. – 2019, 62, ¹ 1. – P. 59–66 (in Ukrainian).
47. Prokip
Volodymyr On Solvability of
the Matrix Equation AX–XB=C over Integer Rings // Modeling, Control and
Information Technologies. – 2019. – ¹ 3. – Ð.55–58. (Scopus)
46. Prokip
V.M. On structure of matrices over a principal ideal domain with respect to
similarity transformation // Proc. Intern. Geom. Center.
– 2019. – V. 12, ¹ 1. – P. 56–69 (in Ukrainian). (Scopus, Q4)
45. Prokip
V.M. On the similarity of matrices AB and BA over a field // Carpathian
Mathematical Publications. – 2018. – V. 10, ¹. 2. – Ñ. 352–359. (Web of Science, Q4)
44. Prokip
V.M. Triangularization of a pair of matrices over the
domain of principal ideals with minimal quadratic polynomials. // Journal of Mathematical Sciences. – 2017. – 222, No. 1 – P. 50–55. (Scopus,
Web of Science,
0.517, Q3)
43. Prokip
V.M. The Structure of Symmetric Solutions of the Matrix Equation AX=B over a
Principal Ideal Domain // Hindawi. International
Journal of Analysis. Volume 2017, Article ID 2867354, 7 pages.
42. Prokip
V.M. On divisibility with a remainder of matrices over a principal ideal domain
// Mat. Metody Fiz.-Mekh. Polya. – 2017, 60, ¹ 2.
P.41–50 (in Ukrainian).
41. Prokip
V.M. The structure of matrices of rank one over the domains of principal ideals
with respect to similarity transformation // Mat.
Metody Fiz.-Mekh. Polya – 2016. –
59, No 3. – P. 68–76 (in Ukrainian).
40. Triangularization
of a pair of matrices over the domain of principal ideals with minimal
quadratic polynomials. (Ukrainian, English) Zbl 1349.15036 Mat. Metody Fiz.-Mekh. Polya
58, No. 1, 42-46 (2015); translation in J. Math. Sci., New York 222, No. 1, 50–55 (2017).
39. A structure of
symmetric solutions of the matrix equation AX
= B over an arbitrary field (in Ukrainian) Proc. Intern. Geom. Center
2016 9(1), 31–37.
38. Simultaneous Triangularization of a Pair of Matrices over a Principal
Ideal Domain with Quadratic Minimal Polynomials // Advances in Linear Algebra
Research. 2015. Novapublisher, New York. P.287–297.
37. On the solvability
of a system of linear equations over the domain of principal ideals. (English.
Ukrainian original) Zbl 1315.15002 Ukr. Math. J.
66, No. 4, 633-637 (2014); translation from Ukr. Mat.
Zh. 66, No. 4, 566–570 (2014).
36. On solutions of the
matrix equation XA0=A1
with prescribed characteristic polynomials (in Ukrainian) Proc. Intern. Geom.
Center. 2014 7(4), 23-33.
35. On normal form with
respect to semiscalar equivalence of polynomial
matrices over a field. (Ukrainian, English) Zbl 1289.15022 Mat. Metody Fiz.-Mekh.
Polya 55, No. 3, 21-26 (2012); translation in J.
Math. Sci., New York 194, No. 2, 149-155 (2013).
34. Diagonalizability
of matrices over a principal ideal domain. (English. Russian original) Zbl 1260.15013 Ukr. Math. J. 64,
No. 2, 316-323 (2012); translation from Ukr. Mat. Zh. 64, No. 2,
283-288 (2012).
33. Canonical form with
respect to semiscalar equivalence for a matrix pencil
with nonsingular first matrix. (English. Russian original) Zbl 1253.15017 Ukr. Math. J.
63, No. 8, 1314-1320 (2012); translation from Ukr.
Mat. Zh. 63, No. 8, 1147-1152 (2011).
32. Diagonalization of
matrices over the domain of principal ideals with minimal polynomial m(λ)=(λ-α)(λ-β), α≠β. (English. Ukrainian original)
Zbl 1281.15013 J. Math. Sci., New York 174, No. 4, 481-485 (2011);
translation from Ukr. Mat. Visn.
7, No. 2, 212-219 (2010).
31. Reduction of a set
of matrices over a principal ideal domain to the Smith normal forms by means of
the same one-sided transformations. (English) Zbl 1215.15012 Olshevsky, Vadim
(ed.) et al., Matrix methods. Theory, algorithms and applications. Dedicated to
the memory of Gene Golub. Based on the 2nd international conference on matrix
methods and operator equations, Moscow, Russia, July 23–27, 2007. Hackensack,
NJ: World Scientific (ISBN 978-981-283-601-4/hbk). 166-174 (2010).
30. About the uniqueness
solution of the matrix polynomial equation A(λ)X(λ)-Y(λ)B(λ) = C(λ). (English)
Zbl 1176.15019 Lobachevskii J.
Math. 29, No. 3, 186-191 (2008).
29. On
triangular unitary divisor of polynomial matrices over factorial domain. (Ukrainian.
English summary) Zbl 1199.15043
Zb. Pr. Inst. Mat. NAN Ukr. 6, No. 2,
35-46 (2009).
28. Common divisors of
matrices over factorial domains. (Ukrainian) Zbl 1108.15018 Mat. Metody Fiz.-Mekh. Polya 48, No. 4,
43-50 (2005).
27. On
similarity of matrices over commutative rings. (English) Zbl 1073.15009 Linear Algebra Appl. 399, 225-233
(2005).
26. On one
class of divisors of polynomial matrices over integral domains. (Ukrainian,
English) Zbl 1073.15511 Ukr. Mat. Zh. 55,
No. 8, 1099-1106 (2003); translation in Ukr. Math. J. 55, No. 8, 1329-1337
(2003).
25. Structure of some sets of
matrices divisors over the principal ideal domain. (Ukrainian) Zbl 1075.15019 Mat. Metody Fiz.-Mekh. Polya
45, No. 3, 14-21 (2002).
24. One class of divisors of
polynomial matrices over a field. (Ukrainian. English
summary) Zbl 1063.65537 Visn.
L’viv. Univ., Ser. Prykl. Mat. Inform. 2002, No. 5, 39-44
(2002).
23. On the
structure of divisors of matrices over a principal ideal domain. (Ukrainian.
English summary) Zbl 1030.15012 Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr.
Tekh. Nauky 2002, No.6, 27-32 (2002).
22. Divisors of polynomial matrices
over a factorial domain. (Ukrainian) Zbl 1098.15503 Mat. Metody Fiz.-Mekh.
Polya 44, No. 4, 22-26 (2001).
21. On multiplicativity of canonical diagonal forms of matrices
over principal ideal domains. II. (English. Ukrainian original) Zbl 0989.15004 Ukr. Math. J. 53, No.2, 312-315 (2001); translation from Ukr.
Mat. Zh. 53, No.2, 274-277 (2001). [For part I see ibid. 47, No. 11, 1581-1585
(1995; Zbl 0888.15004).]
20. Divisors of polynomial matrices
with given canonical diagonal forms. (Ukrainian) Zbl 1053.15006 Mat. Metody Fiz.-Mekh. Polya
43, No. 2, 58-63 (2000).
19. On
solvability of matrix polynomial equations. (Ukrainian. English
summary) Zbl 1048.15013 Visn.
L’viv. Univ., Ser. Prykl. Mat. Inform. 2000, No. 3, 60-64
(2000).
18. Polynomial matrices over a
factorial domain and their factorization with given characteristic polynomials.
(English. Ukrainian original)
Zbl 0942.15009 Ukr. Math. J. 50, No.10, 1644-1647
(1998); translation from Ukr. Mat. Zh. 50, No.10, 1438-1440 (1998).
17. On
common unital divisors of polynomial matrices. (Ukrainian)
Zbl 0924.15010 Mat. Metody Fiz.-Mekh. Polya 40, No.3, 20-24
(1997).
16. On
common unital divisors of polynomial matrices. (Ukrainian)
Zbl 0924.15010 Mat. Metody Fiz.-Mekh. Polya 40, No.3, 20-24
(1997).
15. On the
factorization of polynomial matrices over the domain of principal ideals. (English.
Ukrainian original) Zbl 0940.15015 Ukr. Math. J. 48, No.10, 1628-1632
(1996); translation from Ukr. Mat. Zh. 48, No.10, 1435-1439 (1996).
14. On
factorization of polynomial matrices of two variables over the arbitrary field.
(Ukrainian) Zbl 0926.15010 Dopov. Akad. Nauk Ukr.
1996, No.5, 3-7 (1996).
13. On
multiplicative problem of canonical diagonal forms of matrices over a principal
ideal domain. (Ukrainian) Zbl 0888.15004 Ukr. Mat. Zh. 47,
No.11, 1581-1585 (1995).
12. Parallel factorizations of matrix
polynomials over an arbitrary field. (English. Ukrainian original) Zbl 0868.15012 J. Math. Sci., New York 81, No.6, 3020-3023
(1996); translation from Mat. Metody Fiz.-Mekh. Polya 38,
24-28 (1995).
11. The multiplicativity
of the Smith normal form. (English) Zbl 0824.15009 Linear
Multilinear Algebra 38, No.3, 189-192 (1995).
10. On the
solvability of the Riccati matrix algebraic equation. (English. Russian original) Zbl 0963.93521 Ukr. Math. J. 46, No.11, 1763-1766
(1994); translation from Ukr. Mat. Zh. 46, No.11, 1591-1593 (1994).
09. On common monic divisors having a given canonical diagonal
form for matrix polynomials (with V. Petrichkovich and
F. Pruhnitskii ) Journal of
Mathematical Sciences 79 (6), 1402-1405 translation from Mat. Metody Fiz.-Mekh.
Polya 37, 28-26 (1994).
08. On
common unital divisors of matrix polynomials over an arbitrary field. (English.
Russian original) Zbl 0813.15009 Russ. Acad.
Sci., Sb., Math. 78, No.2, 427-435 (1994); translation from Mat. Sb. 184, No.4, 41-50 (1993).
07. A method for finding a common
linear divisor of the matrix polynomials over an arbitrary field. (English. Ukrainian original) Zbl 0809.15009 Ukr. Math. J. 45, No.8, 1321-1324 (1993);
translation from Ukr. Mat. Zh. 45, No.8, 1181-1183 (1993).
06. On the
uniqueness of the unital divisor of a matrix
polynomial over an arbitrary field. (English. Ukrainian original) Zbl 0854.15004 Ukr. Math. J. 45, No.6, 884-889 (1993); translation from Ukr.
Mat. Zh. 45, No.6, 803-808 (1993).
05. On multiplicativity of canonical diagonal forms of matrices. (English.
Russian original) Zbl 0852.15006 Russ. Math.
36, No.7, 58-60 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992,
No.7(362), 60-62 (1992).
04. On divisibility and one-sided
equivalence of polynomial matrices. (Russian) Zbl 0711.15023 Ukr. Mat. Zh. 42,
No.9, 1213-1219 (1990).
03. On the
separation of the unitary divisor of a rectangular polynomial matrix. (Ukrainian.
Russian summary) Zbl 0706.15013 Ukr. Mat. Zh. 42, No.8, 1089-1094 (1990).
02. Factorization of polynomial
matrices over arbitrary fields. (English. Russian original) Zbl 0613.15010 Ukr. Math. J. 38, 409-412 (1986); translation from Ukr. Mat.
Zh. 38, No.4, 478-483 (1986).
01. On common divisors of matrix
polynomials (with Petrichkovich
V.) Mat. Metody Fiz.-Mekh.
Polya 18, 23-26 (1983).
Some Publications speech at the
conference
41. Prokip Volodymyr. On symmetric solutions of the matrix equation AX
= B over a Bezout domain // ²ííîâàö³éí³
öèôðîâ³ ìåòîäè â ãàëóç³ îñâ³òè òà
äîñë³äæåíü. Çá³ðíèê òåç äîïîâ³äåé.
Áåðåãîâå, 27-28 áåðåçíÿ
2025 ðîêó. P.146-148.
40. Prokip
V. On the uniqueness of solutions of a matrix equation AX - Y B = C over the
ring of integers // 15th Ukraine Algebra Conference, July 8-12, 2025, Lviv, Ukraine. Book of Abstracts: Ivan Franko National
University of Lviv, Ukraine. - 123 p. - P. 84. -
https://xvuac.mmf.com.ua/index.php/abstracts
39. Rostyslava
V. Kolyada, Orest M. Mel'nyk, Volodymyr M. Prokip On solutions of matrix equation A(λ)X(λ)+Y(λ)B(λ)=C(λ)
// 15th Ukraine Algebra Conference, July 8-12, 2025, Lviv,
Ukraine. Book of Abstracts: Ivan Franko National University of Lviv, Ukraine. - 123 p. - P. 61. -
https://xvuac.mmf.com.ua/index.php/abstracts
38. Volodymyr
Prokip On solutions of the matrix equation AX + YB =
C over Bezout domain // Ukraine Mathematics
Conference "At the End of the Year 2025," December 18–19, 2025, Taras Shevchenko National University of Kyiv, Kyiv,
Ukraine, Book of Abstracts. – 133 ñ. – P. 47. –
https://drive.google.com/file/d/1LGUD-of4DUq2LiKYbN233claruwbGIj4/view
37. Prokip
V.M. About square roots of matrices over factorial domains // International
Scientific Conference Algebraic and Geometric Methods of Analysis. Odesa, Ukraine
May 27-30, 2024. Book of Abstracts. P. 102–103.
36. Prokip
V. On solvability of the matrix equation AX+YB=C over principal ideal domains
// V²I International scientific-practical conference "Modeling, control
and information technology" November 7-9 2024, Rivne.–
P. 264–267. https://doi.org/10.31713/MCIT.2024.081
35. Volodymyr
M. Prokip Roth’s theorem and equivalence of matrices
// Ukraine Mathematics Conference "At the End of the Year 2024,"
December 16–18, 2024, Taras Shevchenko National
University of Kyiv, Kyiv, Ukraine, Book of Abstracts. – 92 ñ.
– P. 41. – https://sites.google.com/knu.ua/aey2024/abstracts
34. Prokip
V., Melnyk O., Kolyada R. On divisibility of polynomial matrices over a field
// Current problems of Mechanics and Mathematics – 2023: Collection of
Scientific Papers / Edited by Roman M. Kushnir
(Academician of NAS of Ukraine)
and Volodymyr O. Pelykh
(Corresponding Member of NAS of Ukraine)
[Published Online] // Pidstryhach Institute for
Applied Problems of Mechanics and Mathematics of NAS of Ukraine. – 2023. – 454 p. – Access
by the link: http://iapmm.lviv.ua/mpmm2023/materials/proceedings.mpmm2023.pdf
P. 389–390. – http://iapmm.lviv.ua/mpmm2023/materials/ma10_04.pdf
33. Prokip
V. On canonical form of matrices with minimal polynomial m(λ)=(λ-α)2.
// 14th Ukraine Algebra
Conference, July 3-7, 2023 Sumy,
Ukraine. Book
of Abstracts: Sumy State Pedagogical
University named after A.S. Makarenko,
Sumy, Ukraine. – 150 p. – P.109.
32. Prokip
V.M. On equivalence of polynomial matrices over a field // Ukraine Algebra
Conference "At the End of the Year 2023" December 26 27, 2023, Kyiv, Ukraine.
Book of abstracts: Kyiv, 2023. – 59 p. – P. 46.
31. Prokip
V.M. About solvability of the matrix equation AX= B over Bezout
domains // In International Scientific Conference Algebraic and Geometric
Methods of Analysis. Odesa,
Ukraine May
24-27, 2022. P. 39–40.
30. Prokip
V.M. A note on a minimal solution of the matrix polynomial equation A(λ)X(λ)-Y(λ)B(λ)=C(λ) //
International Algebraic Conference "At the End of the Year" 2022,
December 27–28, 2022, Kyiv, Ukraine, Abstracts: Taras
Shevchenko National University of Kyiv, Kyiv,
Ukraine. – 73
p. – P. 43
29. Prokip
V. Roth's theorems and similarity of matrices // The 13th International
Algebraic Conference in Ukraine.
July 6–9, 2021, Taras Shevchenko National University of Kyiv. Book of
Abstracts: Taras Shevchenko National University of Kyiv, Kyiv, Ukraine. – 94 p.
– Ðåæèì äîñòóïó äî ðåñóðñó: https://bit.ly/3hcTbKe – P. 90.
28. Prokip
V. On the matrix equation AX - YB=C over Bezout domains // International Online
Conference Algebraic and Geometric Methods of Analysis dedicate to the memory
of Yuriy Trokhymchuk (17.03.1928-18.12.2019) (May 25-28, 2021, Odesa, Ukraine).
Book of Abstracts. – 172 p. − P. 122. −
https://www.imath.kiev.ua/~topology/conf/agma2021/contents/agma2021-abstracts.pdf.
27. Prokip
V. A note on semiscalar equivalence of
polynomial matrices // XI Inter.
V.Skorobohatko Math. Conference . L’viv, 2020. Book of Abstracts. P.93.
26. Prokip
V.M. On similarity of families of 2x2 matrices over a field // Book of
abstracts of the International mathematical conference dedicated to the 60th
anniversary of the department of algebra and mathematical logic of Taras
Shevchenko National University of Kyiv, 14-17 July 2020, Kyiv, Ukraine. – 93 p.
– [Electronic resource]. – Access mode: https://bit.ly/2ZIyqMs – P. 68.
25. Prokip
V. On similarity of two families of matrices over a field // International Scientific Conference
Algebraic and Geometric Methods of Analysis, 26-30 may 2020, Odesa, Ukraine. –
131 p. – Access mode: https://www.imath.kiev.ua/~topology/conf/agma2020/agma-2020-abstracts/agma2020-theses.pdf
– P. 55.
24. Prokip
V. On similarity of tuples of matrices over a field // The XII International Algebraic Conference in Ukraine dedicated to the
215th anniversary of V.Bunyakovsky. July 02-06, 2019, Vinnytsia, Ukraine.
Abstracts/ Vinnytsia: Vasyl' Stus Donetsk National University, 2019. – 142 p. –
P. 90–91.
23. Prokip
Volodymyr On the similarity of matrices AB and BA // Modern problems of Mechanics and Mathematics: collection
of scientific papers in 3 vol. / Edited by A.Ì. Samoilenko, R.M. Kushnir
[Electronic resource] // Pidstryhach Institute for Applied Problems of
Mechanics and Mathematics NAS of Ukraine. – 2018. – Vol. 3. – Access mode:
www.iapmm.lviv.ua/mpmm2018. P. 260–261.
22. Prokip
V.M. A note on similarity of matrices // Book of abstracts of the International
Scientific Conference “Algebraic and geometric methods of analysis” (May 30 –
June 4, 2018, Odesa, Ukraine). P. 49. [Electronic resource]. - Access mode: (http://www.imath.kiev.ua/~topology/conf/agma2018)
21. Prokip
V.M. About a structure of solutions of the matrix equation AX - XB = C.
Materials of reports International sciences Conferences "Modern problems
of mathematical modeling, computational methods and information
technologies", Rivne , 2018. P.138-139.
20. Prokip
V.M. On solvability of the matrix equation ÀÕ=XÂ over integral domains // Book
of abstracts of the XI International Algebraic Conference in Ukraine dedicated
to the 75th anniversary of V.V.Kirichenko (July,
2017, Kyiv, Ukraine). P.107.
19. Prokip
V.M. About coexistence of system of matrix equations AX = B ³ BY = A over
commutative rings // PSC–IMFS–13 dedicated to the 125th anniversary Stefana
Banaha. March 30-31, 2017. L'viv, Ukraine. Conference Proceedings. P.65-66. –
[Electronic resource]. – Access mode:
(http://psc-imfs.lpnu.ua/sites/default/files/PSC-13.pdf).
18. Volodymyr Prokip. On
common eigenvectors of two matrices over a principal ideal domain // ̳æíàð. ìàòåì. êîíô. ³ì. Â. ß. Ñêîðîáîãàòüêà, 25 – 28 ñåðïíÿ 2015 ð., Äðîãîáè÷, Óêðà¿íà. Òåçè äîïîâ. – Ñ. 129.
17. Prokip V.M. Solutions of the matrix
equation XA0 = A1
over domains of principal ideals with prescribed characteristic polynomials //
X International Algebraic Conference in Ukraine. Odessa, 2015 ð., Abstracts. – C.91.
16. Volodymyr Prokip.
Solutions of a linear matrix equation XA0
= A1 with prescribed characteristic polynomials // Oblicza Algebry. Ogól-nopolska
Konferencja Naukowa, Kraków, Poland (May 29 – 30, 2015 r.) – Ñ.33. http://algebra.up.krakow.pl http://algebra.up.krakow.pl/abstr-all-strona3.pdf?w=no
15. Prokip V. On common solutions of matrix equations over an elementary divisor domain // International
Algebraic Conference dedicated to 100th anniversary of L.A. Kaluzhnin. Book of Abstracts. Jule
7–12, 2014. Kyiv. P. 69.
14. Prokip V. Normal form with respect to similarity of involutory
matrices over a principal ideal domain // 9th International Algebraic
Con-ference in Ukraine. Abstracts. Jule 8–13, 2013. Lviv. P. 149.
13. Prokip V.M. Note on the Hermite
normal form // International Confe-rence dedicated to the 120-th anniversary of Stefan Banach, Lviv, Ukraine, 17–21 September 2012. Abstracts of Reports. – P. 262.
12. Volodymyr Prokip. Normal form with respect to similarity of a matrix
with minimal polynomial m(λ)=(λ–α)(λ–β),
(α≠β) // International Conference on Algebra dedicated to 100th anniversary of S.M. Chernikov, August 20-26, 2012, Dragomanov
National Pedagogical University, Kiev, Ukraine: Book of abstracts. – Kiev: Institute of Mathematics of UNAS, 2012. – P. 121.
11. Prokip V.M. A structure of GCD of matrices over a
principal ideal domain // International mathematical conference: abstracts of talks. – Mykolayiv: Published by Mykolayiv V.O. Suchomlinsky National University, 2012. – P. 145–146.
10. Prokip V.M. About triangularization of matrices over a principal ideal domain
//
Abstract of 8-th Intern.
Algebraic Conference in Ukraine. Abstract of
talks, Lugansk, 2011. – p.177.
9. Prokip V.M. About
the normal form of linear matrix pencils // III International Conference on matrix methods and operator
equations. Abstracts. Moscow, June 22–25,
2011. P.39.
8. Prokip
V.M. Semiscalar equivalence of polynomial matrices over a field //
International Geom. Conf. «Geometry in Astrahan’ –2009», Astrahan’, 2009. P.40.
7. Prokip V. M. On semiskalar equivalence of polynomial matrices over a field // Abstract of 7–th International
Algebraic Conference in Ukraine. Abstract of
talks, Kharkiv, 2009. P.111–112
6.
Prokip V. On similarity of a matrix and its transpose
// International Conference on Radicals. ICOR-2006. Kyiv,
Ukraine, July 30—August 5, 2006. Abstracts, P. 55.
5. Prokip V. On common divisors of matrices over principal ideal
domain // 5th
International Algebraic Conference in Ukraine. Odessa, July 20-27, 2005.
Abstracts. – Odessa, 2005. – P. 159-160.
4.
Prokip V. About the uniqueness solution of the matrix polynomial equation// 6th
International Algebraic Conference in Ukraine. Kamyanets-Podilsky, July 1-7,
2007. Abstracts. – Kamyanets-Podilsky, 2007. –P. 158-159.
3. Prokip V. On common divisors of matrices over principal ideal domain // 13th
IWMS 2004. August 2004, Poland,
Poznan`. Abstract. – Poznan`, 2004. – P.23. (http://matrix04.amu.edu.pl/pdf/prokip.pdf )
2. Prokip
V.M. On
equivalence relation of polynomial matrices // Algebraic Conference.
Ì. Moskov State Univer., Mech.&Math. Dep., 2004 . P. 256–257.
1. Prokip
V. On a class of divisors of polynomial matrices over integer domains // 4th International Algebraic Conference in
Ukraine Lviv, 4-9 August, 2003, Abstracts. – Lviv, 2003. – P. 177–178.
IMAGE Problem Corner: New Problems
2. Volodymyr Prokip. Problem 55-5: On the Rank of Integral
Matrices. The Bulletin of the International Linear Algebra Society. Issue
Number 55. Fall 2015.
1. Volodymyr Prokip. Problem 51-4: An Adjugate Identity. The
Bulletin of the International Linear Algebra Society. Issue Number 51. Fall 2013.
Phone service: (032) 258 96 23
E-mail: v.prokip@gmail.com vprokip@mail.ru vprokip@ergo.iapmm.lviv.ua
