Generalized equivalence of pairs
of matrices

V. Petrychkovych

Pairs $(A_1,B_1)$ and $(A_2,B_2)$ of matrices over a commutative principal ideal domain $R$ are called generalized eguivalent pairs provided $A_2= UA_1 V_1$, $B_2 = UB_1 V_2$ for some invertible matrices $U,V_1 ,V_2$ over $ R$. The generalized equivalence of pairs of matrices over $ R$ is investigated. In particular, necessary and sufficient conditions are found under which a pair of nonsingular matrices over $ R$ is generalized eguivalent to a pair of diagonal matrices. Some applications of these results are considered.

1. Weierstrass K. Zur Theorie der bilinearen und quadratischen Formen // Monatsh. Akad. Wiss. – Berlin–1867 p. 310–338

2. Kronecker L. Algebraische Reduktion der Scharen bilinearer Formen // Sitz.-Ber. Akad. Wiss. Phys.-math. Klasse. – Berlin–1890 p.763–776

3. Kravchuk M.P. On quadratic forms and linear transformations // Tr. fiz.-mat. viddilu UAN– 1924 V.1 N.3 p.1–91

4. Kravchuk M.P., Goldbaum Ja.S. On equivalence of singular pencil of matrices// Tr. Kyiv. Aviats. in-ty–1936 N.6 p. 5–27

5. Gudyvok P.M. On equivalence of matrices over commutative rings Besk. gruppy i primyk. algebr. struktury,addr Kyiv– 1993 p.431–437

6. Kazimirs'ky P.S., Petrychkovych V.M. On the equivalence of matrix polynomialsTeor. ta prykl. pytannia algebry i dyf. rivn. ,addr Kyiv– 1977 p.61–66

7. Kazimirs'ky P.S. Solution of the problem of the regular factor separation from the matrix polynomial// Ukrain. Mat. Zh.– 1980 V.32 N.4 p.483–498

8. Petrychkovych V.M. Semi-scalar equivalence and the factorization of matrix polynomials// Ukrain. Mat. Zh– 1990 V. 42 N.5 p.644–649

9. Zabavs'ky B.V., Kazimirs'ky P.S. Reduction of a pair of matrices over an adequate ring to a specific triangular form by means of identical unilateral transformation// Ukrain. Mat. Zh.– 1984 V.36 N.2 p.256–258

10. Petrychkovych V.M. On a semi-scalar equivalence and the Smith normal form of polynomial matrices// Mat. Metody i Fiz.-Mekh. Polya– 1987 N.26 p. 13–16

11. Shavarovs'ky B.Z. Semi-scalar equivalence of the polynomial matrices with pairwise distinct roots of their characteristic polynomial// Mat. Metody i Fiz.-Mekh. Polya– 1984 N.19 p.33–37

12. Shavarovs'ky B.Z. Canonical form of polynomial matrix with one elementary divisor to relatively semi-scalar equivalent transformations // Dopovidi Acad. Nauk Ukrain. RSR, Ser. A.– 1988 N.10 p.33–36

13. Feinberg R.B. Equivalence of partitioned matrices// J. Res. Bur. Stand. Sect.–1976 V.80 N.1 p.89–97

14. Ingraham M.N. Rational methods in matrix equations// Bull. Amer. Math. Soc.–1941 V.47 p. 61–70

15. Newman M. On the Smith normal form// J. Res. Bur. Stand. Sect. –1971 V.75 p. 81–84

16. Roth W.E. The equations $AX - YB = C$ and $AX - XB = C$ in matrices // Proc. Amer. Math. Soc.-- 1952 N.3 p. 392--396.

17. Newman M. Multiplicativity of the Smith normal form// Linear and Multilinear Algebra– 1994 V.36 N.3 p.223–224

	Generalized equivalence of pairs of matrices
Author	V. Petrychkovych Pidstryhach Institute of Applied Problems of Mechanicsand Mathematics, 3b Naukova Str., 290601, L'viv, Ukraine
Abstract	Pairs $(A_1,B_1)$ and $(A_2,B_2)$ of matrices over a commutative principal ideal domain $R$ are called generalized eguivalent pairs provided $A_2= UA_1 V_1$, $B_2 = UB_1 V_2$ for some invertible matrices $U,V_1 ,V_2$ over $ R$. The generalized equivalence of pairs of matrices over $ R$ is investigated. In particular, necessary and sufficient conditions are found under which a pair of nonsingular matrices over $ R$ is generalized eguivalent to a pair of diagonal matrices. Some applications of these results are considered.
Keywords	generalized equivalent pairs, matrices, commutative principal ideal domain, invertible matrices, diagonal matrices, necessary and sufficient conditions, applications
DOI	doi:10.30970/ms.8.2.147-152
Reference	1. Weierstrass K. Zur Theorie der bilinearen und quadratischen Formen // Monatsh. Akad. Wiss. – Berlin–1867 p. 310–338 2. Kronecker L. Algebraische Reduktion der Scharen bilinearer Formen // Sitz.-Ber. Akad. Wiss. Phys.-math. Klasse. – Berlin–1890 p.763–776 3. Kravchuk M.P. On quadratic forms and linear transformations // Tr. fiz.-mat. viddilu UAN– 1924 V.1 N.3 p.1–91 4. Kravchuk M.P., Goldbaum Ja.S. On equivalence of singular pencil of matrices// Tr. Kyiv. Aviats. in-ty–1936 N.6 p. 5–27 5. Gudyvok P.M. On equivalence of matrices over commutative rings Besk. gruppy i primyk. algebr. struktury,addr Kyiv– 1993 p.431–437 6. Kazimirs'ky P.S., Petrychkovych V.M. On the equivalence of matrix polynomialsTeor. ta prykl. pytannia algebry i dyf. rivn. ,addr Kyiv– 1977 p.61–66 7. Kazimirs'ky P.S. Solution of the problem of the regular factor separation from the matrix polynomial// Ukrain. Mat. Zh.– 1980 V.32 N.4 p.483–498 8. Petrychkovych V.M. Semi-scalar equivalence and the factorization of matrix polynomials// Ukrain. Mat. Zh– 1990 V. 42 N.5 p.644–649 9. Zabavs'ky B.V., Kazimirs'ky P.S. Reduction of a pair of matrices over an adequate ring to a specific triangular form by means of identical unilateral transformation// Ukrain. Mat. Zh.– 1984 V.36 N.2 p.256–258 10. Petrychkovych V.M. On a semi-scalar equivalence and the Smith normal form of polynomial matrices// Mat. Metody i Fiz.-Mekh. Polya– 1987 N.26 p. 13–16 11. Shavarovs'ky B.Z. Semi-scalar equivalence of the polynomial matrices with pairwise distinct roots of their characteristic polynomial// Mat. Metody i Fiz.-Mekh. Polya– 1984 N.19 p.33–37 12. Shavarovs'ky B.Z. Canonical form of polynomial matrix with one elementary divisor to relatively semi-scalar equivalent transformations // Dopovidi Acad. Nauk Ukrain. RSR, Ser. A.– 1988 N.10 p.33–36 13. Feinberg R.B. Equivalence of partitioned matrices// J. Res. Bur. Stand. Sect.–1976 V.80 N.1 p.89–97 14. Ingraham M.N. Rational methods in matrix equations// Bull. Amer. Math. Soc.–1941 V.47 p. 61–70 15. Newman M. On the Smith normal form// J. Res. Bur. Stand. Sect. –1971 V.75 p. 81–84 16. Roth W.E. The equations $AX - YB = C$ and $AX - XB = C$ in matrices // Proc. Amer. Math. Soc.-- 1952 N.3 p. 392--396. 17. Newman M. Multiplicativity of the Smith normal form// Linear and Multilinear Algebra– 1994 V.36 N.3 p.223–224
Pages	147-152
Volume	8
Issue	2
Year	1997
Journal	Matematychni Studii
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Table of content of issue	html

Generalized equivalence of pairs of matrices