Prokip Volodymyr Mykhajlovych
Education:
Scientific degree: Candidate of Science (Ph.D.),
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),
Position: Senior Researcher Worker (April 1997 - present).
In Institute: April 1987 - present.
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
54. Prokip V. M. A note on semiscalar
equivalence of polynomial matrices // Electronic Journal of Linear Algebra. –
2022. – v. 38. – P. 195–203.
53. Prokip V. Equivalence of Polynomial
Matrices over a Field // arXiv preprint arXiv: 2003.05041, 2020.
52. Prokip V. M. Equivalence of Polynomial Matrices
over a Field // Hot Topics in Linear Algebra. Chapter 6. 2020. Ð. 205–232.
51. 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.
50. 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
49. 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
48. 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.
47. 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).
46. 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.
45. Prokip
V.M. On structure of matrices over a principal ideal
domain with respect to similarity transformation // Proc. Intern.
44. 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.
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.,
39. A structure of
symmetric solutions of the matrix equation AX
= B over an arbitrary field (in Ukrainian) Proc. Intern.
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.,
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
29. Prokip V. Roth's theorems and similarity of matrices // The
13th International Algebraic Conference in
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
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
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