Shavarovskii Bogdan Zenoviyovych

Education: Drohobych Ivan Franko State Pedagogikal University (specialty mathematics and physics, 1973), Postgraduate study at Pidstryhach IAPMM) (01.01.06 algebra and number theory, 1979 )

Scientific degree: Doctor of sciences in physics and mathematics (doctoral thesis "Reduction and decomposition of matrices over polynomial rings", 2013 ., 01.01.06 algebra and number theory.)

Scientific title: Senior Research Fellow (1995)

Position: Acting Leading Research Fellow (from 2013 .)

In Institute: from 1974

Research interests: linear algebra, matrix theory, invariant theory, matrix equations, computational methods of linear algebra

Direction of scientific researches: Reduction of matrix over rings and reduction of sets of matrices over fields: search invariants, including their complete systems; build to simple form; special types of transformation. Decomposition of matrices over ring: factorization of matrices into a product of regular factors of lower degrees. Solving polynomial equations of higher degrees and systems of linear equations bilateral matrix.

Main scientific results:


  1. Developed and applied a method of semiscalar equivalence of polynomial matrices to the similarity problem of finite sets of numerical matrices.
  2. The problem of reduction of rather wide classes of polynomial matrices to direct sum of summands has been solved. This enabled to reduce complex problems of semiscalar equivalence and similarity of matrices to analogous problems for matrices of lower dimension.
  3. For some classes of finite sets of numerical matrices constructed canonical and related to them forms and found systems of invariants with respect to similarity transformations.
  4. Investigated the equivalence and similarity matrices by different transformations using the matrices of special block structure.
  5. The decomposabiliti into factors (in particular linear) of some classes of matrix polynomials with elementary divisors of degrees more than two has been proved, the classes including the majority of known now classes of factorable matrix polynomials.
  6. The results obtained have been applied to the problems of construction the canonical forms matrix polynomials relating to similarity transformation and also to solution of various types of matrix algebraic equations (polynomial higher degrees, linear two-sided, Riccati equations).


Major publications:


  1. Shavarovskii B.Z. On invariants and canonical form of matrices of second order with respect to semiscalar equivalence // BASM. 2016. n.3 (82). P. 1223.
  2. Shavarovskii B. Z. The block-diagonal similarity and semisalar equivalence of polynomial matrices // Mathematical methods and physico-mechanical fields. 2015. 58, No 1. P. 2941.
  3. Shavarovskii B. Z. The semiscalar equivalence and quasi-diagonal similarity of matrices // Mathematical methods and physico-mechanical fields. 2015. 58, No 4. P. 1526.
  4. Shavarovskii B. Z. On nonsingular solutions of one type of systems of matrix equations // Applied problems of mechanics and mathematics. 2015. Issue. 13. P. 1117,
  5. Shavarovskii B. Z. Semisalar equivalence of polynomial matrices and triangular similarity of numerical matrices // Applied problems of mechanics and mathematics. 2013. Issue. 11. P. 6369.
  6. Shavarovskii B.Z. Canonical form of polynomial matrices with all identical elementary divisors // Ukr. Math. J. 2012. 64, No. 2. P. 282297.
  7. Shavarovskii B.Z. Factorability of polynomial matrices depending on the powers of their elementary divisors // Zh Vychisl. Mat. Mat Fiz 2012. 52, 7. P. 11711184.
  8. Shavarovskii B. Z. On a triangular form of polynomial matrix and its invariants with respect to semiscalar equivalence // Mathematical methods and physico-mechanical fields. 2012. 55, No 3. P. 720.
  9. Shavarovskii B. Z. Canonical form for pair of matrices, in which one matrix has pairwise relatively prime elementary divisors of degree at most two // Applied problems of mechanics and mathematics. 2012. Issue. 10. P. 6270).
  10. Shavarovskii B. Z On the similarity of matrix polynomials of simple structure // Applied problems of mechanics and mathematics. 2011. Issue. 9. P. 7481.
  11. Shavarovskii B.Z. Decomposability of matrix polynomials with commuting coefficients into a product of linear factors // Ukr. Math. J. 2011. 62, No. 8. P. 12951306.
  12. Shavarovskii B.Z. Similarity Transformations of Decomposable Matrix Polynomials and Related Questions // Computational Mathematics and Mathematical Physics. Vol. 49, No. 9, 2009. P. 14691482.
  13. Shavarovskii B.Z. Classes of Factorable and Unfactorable Polynomial Matrices and Their Relative Position // Doklady Mathematics . 2007 75, .2. . 318321.
  14. Shavarovskii B.Z. On the Similarity of Matrices of Even Order. // Mathematical Notes. 81, no. 3. 2007. P. 392407.
  15. Shavarovskii B.Z. Finding a Complete Set of Solutions or Proving Unsolvability for Certain Classes of Matrix Polynomial Equations with Commuting Coefficients // Computational Mathematics and Mathematical Physics. 2007. 47, . 12. . 19021911.
  16. Shavarovskii B.Z. Multiplicative Structure of a Polynomial Matrix Depending on Its Elementary Divisors// Doklady Mathematics , 2007. 76, 3. . 843847.
  17. Shavarovskii B.Z. Solvability of Matrix Equations in Rings of Quasi-Diagonal Matrices and Similarity of Matrix Polynomials // Computational Mathematics and Mathematical Physics. 46. . 8. 2006. P. 12831292.
  18. Shavarovskii B.Z. On Some Tame and Wild Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices // Mathematical Notes. 76, no. 1. 2004. P. 111123.
  19. Shavarovskii B.Z. Solving a System of Linear  Homogeneous Two-Sided Matrix Equations in Two Unknowns. // Computational Mathematics and Mathematical Physics. 44. 11. 2004. P. 18531866.
  20. Shavarovskii B.Z. A Complete System of Invariants with Respect to Similarity for Pairs of Matrices of Even Order with Characteristic Matrix of Special Smith form // Mathematical Notes. 73, no. 6. 2003. P. 871888.
  21. Shavarovskii B.Z. On a Method for Reducing a Pair of Matrices to Block Diagonal Form by a Similarity Transformation // Computational Mathematics and Mathematical Physics . 43. 9. 2003. P. 12311240.
  22. Shavarovskii B.Z. Factorable polynomial matrices // Mathematical Notes. 2000. 68, 4. . 507-518
  23. Shavarovskii B.Z. On Semiscalar and Quasidiagonal Equivalences of Matrices // Ukr. Math. J. 2000. 52, No. 10. P. 16381643.
  24. Shavarovskii B.Z. Decomposition of polynomial matrices into a direct sum of triangular summands // Ukr. Math. J. 1999. 51, No. 8. P. 12911295.
  25. Shavarovskii B.Z. On decomposition of polynomial matrices into direct sum of matrices of smaller order // Reports of the Academy of Sciences of Ukraine. 1999. 4. P. 4953.
  26. Shavarovskii B.Z., Reduction of matrices of equivalent transformations and similarity transformations // Mathematical Notes. 1998. 64, 5. P. 663673.

List of Conference Proceedings:

  1. Shavarovskii B.Z. Improved triangular form of the matrix and its invariants // 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.123.
  2. B.Z. Shavarovskii. When polynomial matrices of second order semiscalarly are equivalent? // X International Algebraic Conference in Ukraine dedicated to the 70th anniversary of Yu. A. Drozd. August 20 27, 2015. Odessa, Ukraine. Abstracts. P. 100.
  3. B.Z. Shavarovskii. Quasi-diagonal similarity and semiscalar equivalence of matrices // International Algebraic Conference dedicated to 100th anniversary of L.A. Kaluzhnin. Book of Abstracts. Jule 712, 2014. Kyiv. P. 76 77.
  4. B.Z. Shavarovskii. Reduction of polynomial matrices by semiscalar equivalent transformation // 9th International Algebraic Conference in Ukraine. July 813, 2013, Lviv, Ukraine. Book of abstracts. P. 172.
  5. Bogdan Shavarovskii On the canonical form for a certain class of matrices with respect to similarity // International Conference dedicated to the 120th anniversary of Stefan Banach, Lviv, Ukraine, 1721 September 2012. Abstracts of Reports. P. 264.
  6. B. Shavarovskii. On similarity of pairs of matrices, in which one matrix has only one elementary divisor // International mathematical conference: abstracts of talks. Mykolayiv: Published by Mykolayiv V.O. Suchomlinsky National University, 2012. P. 134.
  7. B. Shavarovskii Canonical forms for matrices under semiscalar equivalence // Book of abstracts of the 8th International Algebraic Conference in Ukraine (July 5 12, 2011, Lugansk Taras Shevchenko National University, Ukraine), P. 181.
  8. B. Shavarovskii. On the invariants of polynomial matrices with respect to semiscalar equivalence // 7 International Algebraic Conference in Ukraine: abstracts of talks ( 18 23 August, 2009, Kharkov) Kiev, 2009. 168 p. P. 127 128.
  9. Bogdan Shavarovskii. Solving a System of Linear Two-Sided Matrix Equations // 2009, .
  10. Shavarovskii B. On the class of factorable polynomial matrices // 6th International Algebraic Conference in Ukraine. Kamyanets-Podilsky, July 1-7, 2007. Abstracts. Kamyanets-Podilsky, 2007. P. 187-188.
  11. Shavarovskii B.Z. On similarity of some collection of matrices // 5th International Algebraic Conference in Ukraine. Odessa, July 20-27, 2005. Abstracts. Odessa, 2005. P. 186-187.

Phone service: (032) 258 96 23