Generalized equivalence of collections of matrices and common divisors of matrices

The collections (A1, ..., Ak) and (B1, ..., Bk) of matrices over an adequate ring are called generalized equivalent if Ai = UBiVi for some invertible matrices U and Vi , i = 1, ..., k. Some conditions are established under which the finite collection consisting of the matrix and its the divisor...

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Published in:	Algebra and Discrete Mathematics
Date:	2004
Main Author:	Petrychkovych, V.M.
Format:	Article
Language:	English
Published:	Інститут прикладної математики і механіки НАН України 2004
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Cite this:	Generalized equivalence of collections of matrices and common divisors of matrices / V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 84–91. — Бібліогр.: 14 назв. — англ.

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spelling	Petrychkovych, V.M. 2019-06-18T13:30:25Z 2019-06-18T13:30:25Z 2004 Generalized equivalence of collections of matrices and common divisors of matrices / V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 84–91. — Бібліогр.: 14 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 15A33, 15A21, 15A23. https://nasplib.isofts.kiev.ua/handle/123456789/156417 The collections (A1, ..., Ak) and (B1, ..., Bk) of matrices over an adequate ring are called generalized equivalent if Ai = UBiVi for some invertible matrices U and Vi , i = 1, ..., k. Some conditions are established under which the finite collection consisting of the matrix and its the divisors is generalized equivalent to the collection of the matrices of the triangular and diagonal forms. By using these forms the common divisors of matrices is described. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Generalized equivalence of collections of matrices and common divisors of matrices Article published earlier
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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title	Generalized equivalence of collections of matrices and common divisors of matrices
spellingShingle	Generalized equivalence of collections of matrices and common divisors of matrices Petrychkovych, V.M.
title_short	Generalized equivalence of collections of matrices and common divisors of matrices
title_full	Generalized equivalence of collections of matrices and common divisors of matrices
title_fullStr	Generalized equivalence of collections of matrices and common divisors of matrices
title_full_unstemmed	Generalized equivalence of collections of matrices and common divisors of matrices
title_sort	generalized equivalence of collections of matrices and common divisors of matrices
author	Petrychkovych, V.M.
author_facet	Petrychkovych, V.M.
publishDate	2004
language	English
container_title	Algebra and Discrete Mathematics
publisher	Інститут прикладної математики і механіки НАН України
format	Article
description	The collections (A1, ..., Ak) and (B1, ..., Bk) of matrices over an adequate ring are called generalized equivalent if Ai = UBiVi for some invertible matrices U and Vi , i = 1, ..., k. Some conditions are established under which the finite collection consisting of the matrix and its the divisors is generalized equivalent to the collection of the matrices of the triangular and diagonal forms. By using these forms the common divisors of matrices is described.
issn	1726-3255
url	https://nasplib.isofts.kiev.ua/handle/123456789/156417
citation_txt	Generalized equivalence of collections of matrices and common divisors of matrices / V.M. Petrychkovych // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 2. — С. 84–91. — Бібліогр.: 14 назв. — англ.
work_keys_str_mv	AT petrychkovychvm generalizedequivalenceofcollectionsofmatricesandcommondivisorsofmatrices
first_indexed	2025-11-24T21:47:56Z
last_indexed	2025-11-24T21:47:56Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2004). pp. 84 – 91 c© Journal “Algebra and Discrete Mathematics” Generalized equivalence of collections of matrices and common divisors of matrices Vasyl‘ M. Petrychkovych Communicated by M. Ya. Komarnytskyj Abstract. The collections (A1, ..., Ak) and (B1, ..., Bk) of matrices over an adequate ring are called generalized equivalent if Ai = UBiVi for some invertible matrices U and Vi, i = 1, ..., k. Some conditions are established under which the finite collection consisting of the matrix and its the divisors is generalized equivalent to the collection of the matrices of the triangular and diagonal forms. By using these forms the common divisors of matrices is described. 1. Introduction Let R be a commutative ring with 1 6= 0. We denote by M(m, n, R) and M(n, R) the set of m× n matrices and the ring of n× n matrices over R respectively. The collections of matrices (A1, ..., Ak), (B1, ..., Bk), Ai, Bi ∈ M(m, n, R), i = 1, . . . , k are called equivalent if Ai = UBiV, i = 1, ..., k for some invertible matrices U ∈ GL(m, R) and V ∈ GL(n, R). Definition 1. The collections of matrices (A1, ..., Ak), (B1, ..., Bk), Ai, Bi ∈ M(m, ni, R), i = 1, . . . , k 2000 Mathematics Subject Classification: 15A33, 15A21, 15A23. Key words and phrases: collection of matrices, generalized equivalence, canon- ical diagonal form, common divisors. Jo u rn al A lg eb ra D is cr et e M at h .V. M. Petrychkovych 85 are called generalized equivalent if Ai = UBiVi, i = 1, ..., k for some matrices U ∈ GL(m, R) and Vi ∈ GL(ni, R). It is known that the problem of equivalence of the collections of ma- trices is solved at most for pairs of matrices over the fields. This problem is wild even for pairs of matrices over the rings [1]. The reducibility of the finite collections of matrices over polynomial rings and pairs of matrices over the principal ideal rings and other rings by means of gen- eralized equivalent transformations to the triangular and diagonal forms and their applications is considered in [2-7]. V.Dlab and C.M.Ringel [8] have established the canonical form of the pairs of complex matri- ces (A1, A2) with respect to the transformation (A1, A2)(Q, P1, P2) = (QA1P −1 1 , QA2P −1 2 ), where Q is a complex invertible matrix P1 and P2 are real invertible matrices. A standard form of pairs of matrices over a principal ideal ring and an adequate ring with respect to generalized equivalence is established in [9,10]. In this paper we give conditions under which a finite collection consisting of the matrix and its divisors over an adequate ring is generalized equivalent to the collection of the matrices of the triangular and diagonal forms. By using these forms we describe the common divisors of matrices. 2. Generalized equivalence of collections of matrices From now on R will denote an adequate ring, i.e. R be domain of integrity in which every finitely generated ideal is principal and for every a, b ∈ R with a 6= 0, a can be represented as a = cd, where (c, b) = 1 and (di, b) 6= 1 for any non-unit factor di of d [11]. Notice that more general concept of the adequate rings was introduced in [12,13]. We shall denote by DA the canonical diagonal form (Smith normal form) of the matrix A ∈ M(m, n, R), i.e DA = UAV = diag(µ1, . . . , µr, 0, . . . , 0), µr 6= 0, µ1\|µ2\| · · · \|µr for some matrices U ∈ GL(m, R) and V ∈ GL(n, R). Let the matrix Ci ∈ M(m, R) with the canonical diagonal form DCi = Φ is a left divisor of the matrix A ∈ M(m, n, R), m ≤ n, i.e A = CiAi, Ai ∈ M(m, n, R). (1) Jo u rn al A lg eb ra D is cr et e M at h .86Generalized equivalence of collections of matrices and... Then to the factorization (1) of the matrix A there corresponds a factor- ization of its canonical diagonal form DA DA = ΦΨ = diag(ϕ1, . . . , ϕs, 0, . . . , 0)diag(ψ1, . . . , ψt, 0, . . . , 0), ϕ1\|ϕ2\| · · · \|ϕr, D Ci = Φ. Theorem 1. Let A ∈ M(m, n, R), m ≤ n. Suppose that the matrices Ci ∈ M(m, R), i = 1, 2, . . . with canonical diagonal forms DCi = Φ are the left divisors of the matrix A. Then every finite collection of matrices (A, C1, ..., Ck) is generalized equivalent to the collection of the diagonal matrices (DA, DC1 , . . . , DCk) if and only if the matrices Ai in (1) are equivalent to Ψ in (2) for every i = 1, . . . , k. Proof. Let the collection of matrices (A, C1, ..., Ck) is such that the ma- trices Ai in (1) are equivalent to Ψ in (2) for every i = 1, . . . , k. The proof is by induction with respect to k. For k = 1 the proof follows from Theorem 2 in [7]. We assume that Theorem 1 is valid for k − 1 and prove it for an arbitrary k. According to the induction assumption the collection of the matrices (A, C1, ..., Ck−1) is generalized equivalent to the collection of the diagonal matrices (DA, DC1 , . . . , DCk−1), i.e. UAV = DA, UCiVi = DCi for some matrices U, Vi ∈ GL(m, R), i = 1, . . . , k − 1 and V ∈ GL(n, R). Then the collection of the matrices (A, C1, ..., Ck−1, Ck) is generalized equiv- alent to the collection of the matrices (DA, DC1 , . . . , DCk−1 , C̃k), where C̃k = UCk. Since A = CkAk then DA = C̃kÃk, where DA = UAV. Therefore the pair of matrices (DA, C̃k) is generalized equivalent to the pair (DA, DCk), i.e. ŨDAṼ = DA, Ũ C̃kṼk = DCk , where Ũ , Ṽk ∈ GL(m, R), Ṽ ∈ GL(n, R). The matrix Ũ has the form Ũ = ‖uij‖ m 1 , where uij = { µi µj u′ ij , if i, j = 1, . . . , r, i > j, 0, if i = r + 1, . . . , m, j = 1, . . . , r. Therefore ŨΦ = ΦS for some matrix S ∈ GL(m, R). From this we obtain that the collection of the matrices (DA, DC1 , . . . , DCk−1 , C̃k) is generalized equivalent to the collection (DA, DC1 , . . . , DCk−1 , DCk) of the diagonal matrices. Now suppose that the collection of matrices (A, C1, ..., Ck) is gener- alized equivalent to the collection (DA, DC1 , . . . , DCk) of the diagonal matrices. Then it is easy verified that the matrices Ai in (1) are equiva- lent to Ψ in (2) for every i = 1, . . . , k. This completes the proof. Corollary 1. Suppose that the canonical diagonal form DA of the matrix A ∈ M(m, n, R), m ≤ n, can be represented as DA = ΦΨ = diag(ϕ1, . . . , ϕm)diag(ψ1, . . . , ψm), ϕ1\|ϕ2\| · · · \|ϕm, Jo u rn al A lg eb ra D is cr et e M at h .V. M. Petrychkovych 87 Φ ∈ M(m, R), Ψ ∈ M(m, n, R). If ( ϕi ϕj , (ψi, ψj) ) = 1, i, j = 1, . . . , m, i > j, then every finite collection (A, C1, ..., Ck) consisting of the matrix A and its the divisors Ci with canonical diagonal forms DCi = Φ, i = 1, . . . , k is generalized equivalent to the collection (DA, DC1 , . . . , DCk) of the di- agonal matrices. Theorem 2. Let A ∈ M(m, n1, R), B ∈ M(m, n2, R), m ≤ n1, n2, and C ∈ M(m, R), DC = Φ = diag(ϕ1, . . . , ϕs, 0, . . . , 0). Suppose that the matrix C is a common left divisor of the matrices A and B, i.e. A = CA1, B = CB1, (2) and thus DA = ΦΨ, DB = ΦΛ, (3) where Ψ = diag(ψ1, . . . , ψt, 0, . . . , 0), Λ = diag(λ1, . . . , λq, 0, . . . , 0). If the matrix A1 is equivalent to Ψ then the three of matrices (A, B, C) is generalized equivalent to the three (DA, ΦT, DC), where T is lower triangular matrix. Proof. By Theorem 1 the pair of matrices (A, C) is generalized equivalent to the pair of the diagonal matrices (DA, Φ), i.e. UAV = DA, UCW = DC = Φ for some matrices U, W ∈ GL(m, R) and V ∈ GL(n, R). Then the three of matrices (A, B, C) is generalized equivalent to the three ma- trix (DA, B̃ = ΦB̃1, Φ), where B̃ = UB, B̃1 = W−1B1. By Lemma 1 in [9,10] there exist an upper unitriangular matrix U1 ∈ GL(m, R) and an invertible matrix S ∈ GL(n, R) such that U1B̃S = TB , where TB is the lower triangular matrix with the principal diagonal DB. Since U1Φ = ΦŨ1 for some matrix Ũ1 ∈ GL(m, R) then TB = ΦT , where T is the lower triangular matrix. Then U1D AV1 = DA, U1ΦW1 = Φ for some matrices U1, W1 ∈ GL(m, R) and V1 ∈ GL(n1, R). This proves the theorem. Remark 1. If the matrix B1 in (3) is equivalent to Λ in (4) then the three of matrices (A, B, C) is generalized equivalent to the three (ΦT̃ , DB, DC), where T̃ is lower triangular matrix. Jo u rn al A lg eb ra D is cr et e M at h .88Generalized equivalence of collections of matrices and... 3. Common divisors of matrices Let A ∈ M(m, n1, R), B ∈ M(m, n2, R), m ≤ n1, n2. Then the pair of matrices (A, B) is generalized equivalent to the pair (DA, TB = TDB), i.e. UAV1 = DA, UBV2 = TB, (4) where U ∈ GL(m, R), V1 ∈ GL(n1, R), V2 ∈ GL(n2, R), and T is the lower triangular matrix [10]. Theorem 3. Let A ∈ M(m, n1, R), B ∈ M(m, n2, R), m ≤ n1, n2, and DA = diag(µ1, . . . , µr, 0, . . . , 0). Suppose that d-matrix Φ = diag(ϕ1, . . . , ϕs, 0, . . . , 0), ϕ1\|ϕ2\| · · · \|ϕs, Φ ∈ M(m, R) is common divisor of the canonical diagonal forms DA and DB of matrices A and B, i.e. DA = ΦΨ, DB = ΦΛ. (5) Then there exists a common left divisor C with the canonical diagonal form DC = Φ of matrices A and B, i.e. A = CA1, B = CB1, (6) where the matrix A1 is equivalent to Ψ if and only if Φ is the common left divisor of matrices DA and TB. All common left divisors Ci with the canonical diagonal form DCi = Φ of matrices A and B up to associates have the form Ci = U−1 i Φ, where matrices Ui are given by (5). Proof. Let a matrix C ∈ M(m, R) is the common left divisor of the matrices A and B, i.e. the relation (7) holds. By Theorem 2 the pair of matrices (A, B) is generalized equivalent to the pair (DA, T̃B = ΦT̃ ), i.e. ŨAṼ1 = DA, ŨBṼ2 = T̃B (7) for some matrices Ũ ∈ GL(m, R), Ṽ1 ∈ GL(n1, R), Ṽ2 ∈ GL(n2, R). We now must show that the matrix Φ is the left divisor of the matrix TB = UBV2 in (5). For the matrices U in (5) and Ũ in (8) we have U = HŨ1, where matrix H has the form H = ‖hij‖ m 1 , hij = { µi µj h′ ij , if i, j = 1, . . . , r, i > j, 0, if i = r + 1, . . . , m, j = 1, . . . , r. Then TB = HT̃BV3, where V3 = Ṽ −1 2 V2. Since DA = ΦΨ, then HΦ = ΦH̃ for some matrix H̃ ∈ GL(m, R). Thus TB = ΦT. Jo u rn al A lg eb ra D is cr et e M at h .V. M. Petrychkovych 89 Now let Φ is the common left divisor of the matrices DA and TB in (5). Then every matrix U−1 i Φ = Ci, where Ui is given by (5), is the common left divisor of the matrices A and B. If the matrix C ∈ M(m, R) is the common left divisor of the matrices A and B, i.e. the relation (7) holds, then by Theorem 2 there exist the matrices U, V3 ∈ GL(m, R), and V1 ∈ GL(n1, R), V2 ∈ GL(n2, R) such that UAV1 = DA, UBV2 = TB, UCV3 = Φ. Therefore C = U−1Φ. This completes the proof of the theorem. Remark 2. If in (7) the matrix B1 is equivalent to Λ in (6) then there exists a common left divisor C with the canonical diagonal form DC = Φ of matrices A and B if and only if Φ is the common left divisor of matrices TA and DB, where TA = UAV1, DB = UBV2 for some matrices U ∈ GL(m, R), V1 ∈ GL(n1, R) and V2 ∈ GL(n2, R). Corollary 2. Suppose that the canonical diagonal forms DA and DB of the matrices A ∈ M(m, n1, R) and B ∈ M(m, n2, R), m ≤ n1, n2, can be represented as DA = diag(ϕ1, . . . , ϕm)diag(ψ1, . . . , ψm), DB = diag(ϕ1, . . . , ϕm)diag(λ1, . . . , λm), ϕ1\|ϕ2\| · · · \|ϕm. If ( ϕi ϕj , (ψi, ψj) ) = 1, i, j = 1, . . . , m, i > j, or ( ϕi ϕj , (λi, λj) ) = 1, i, j = 1, . . . , m, i > j then there exists common left divisor C with the canonical diagonal form DC = Φ = diag(ϕ1, . . . , ϕm) of matrices A and B if and only if Φ is common left divisor of matrices DA and TB or TA and DB. Corollary 3. Let a pair (A, B) of matrices A ∈ M(m, R), det A 6= 0 and B ∈ M(m, n, R) is generalized equivalent to the pair (DA, DB) of the diagonal matrices, i.e. the matrix (adjA)B is equivalent to the matrix (adjDA)DB [10]. Then there exists common left divisor C with the canonical diagonal form DC = Φ of matrices A and B if and only if Φ is common divisor the canonical diagonal forms DA and DB of matrices A and B. Let A, B ∈ M(m, R). The matrices A and B are called left relatively prime if every factorization A = CiAi, B = CiBi, Ci ∈ M(m, R) of the matrices A and B yields Ci ∈ GL(m, R). Jo u rn al A lg eb ra D is cr et e M at h .90Generalized equivalence of collections of matrices and... Theorem 4. Let A, B ∈ M(m, R) and DA = diag(1, . . . , 1, ϕm), DB = diag(ψ1, . . . , ψm). Then the matrices A and B are left relatively prime if and only if the pair of matrices (A, B) is generalized equivalent to the pair (DA, TB), where TB = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ψ1 0 · · · 0 0 0 ψ2 · · · 0 0 · · · · · · · · · · · · · · · tψ1 0 . . . 0 ψm ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ (8) and t = { 0, if (ϕm, ψm) = 1, 1, otherwise. Proof. Suppose that the matrices A and B are left relatively prime. By Theorem 1 in [10] the pair of matrices (A, B) is generalized equivalent to the pair (DA, TB 1 ), where TB 1 = ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ψ1 0 · · · 0 0 0 ψ2 · · · 0 0 · · · · · · · · · · · · · · · 0 0 · · · ψm−1 0 t1ψ1 t2ψ2 . . . tm−1ψm−1 ψm ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ ∥ . Then the matrices DA and TB 1 are left relatively prime. Therefore (ϕm, (t1ψ1, t2ψ2, . . . , tm−1ψm−1, ψm)) = 1. By Theorem 3 in [10] the pair of matrices (DA, TB 1 ) is generalized equiv- alent to the pair (DA, TB), where the matrix TB has the form (9). Now let the pair of matrices (A, B) is generalized equivalent to the pair (DA, TB), where the matrix TB has the form (9). Then [14] the common left divisors Ci, i = 1, 2 . . . of matrices DA and TB up to right associates have the form Ci = diag(1, . . . , 1, ϕi), where ϕi are the divisors of ϕm. Taking into account the form of the matrices DA and TB we have that ϕi = 1. Therefore the matrices DA and TB are left relatively prime and thus the matrices A and B are left relatively prime. This proves the theorem. References [1] P. M. Gudyvok, On equivalence of matrices over a commutative rings, Besk. gruppy i primyk. algebr. struktury, Kyiv, 1993, pp. 431–437 (in Russian). [2] P.S. Kazimirs’kij, V.M.Petrychkovych, Equivalence of polynomial matrices, Teor. ta prykl. pytannia algebry i dyf. rivnian. Naukova Dumka, Kyiv, 1977, pp. 61-66 (in Ukrainian). Jo u rn al A lg eb ra D is cr et e M at h .V. M. Petrychkovych 91 [3] V. M. Petrychkovych, Semiscalar equivalence and the Smith normal form of polynomial matrices, Mat. Metody Fiz.-Mekh. Polya, N.26, 1987, pp. 13-16 (in Ukrainian). [4] V. M.Petrychkovych, Semiscalar equivalence and the factorization of polynomial matrices, Ukr. Math. Zh., 42, N. 5, 1990, pp. 644-649 (in Ukrainian). [5] B. V.Zabavs’kij, P. S. Kazimirs‘kij, Reduction of a pair of matrices over an ade- quate ring to special triangular form by application of identical unilateral trans- formations, Ukr. Math. Zh., 36, N. 2, 1984, pp. 256–258 (in Ukrainian). [6] V. M.Petrychkovych, A criterion of diagonalizability of a pair of matrices over a ring of principal ideals by common row and separate column transformations, Ukr. Math. Zh. 49, N. 6, 1997, pp. 860-862 (in Ukrainian). [7] V. M.Petrychkovych, Reducibility of pairs of matrices by generalized equivalent transformations to triangular and diagonal forms and their applications, Mat. Metody Fiz.-Mekh. Polya, 43, N. 2, 2000, pp. 15-22 (in Ukrainian). [8] V.Dlab, C. M.F.Ringel, Canonical forms of pairs of complex matrices, Linear Algebra Appl., 147, 1991, pp. 387-410. [9] V. Petrychkovych, Generalized equivalence of pairs of matrices, Linear and Mul- tilinear Algebra, 48, 2000, pp. 179-188. [10] V. M. Petrychkovych, Standard forms of pair of matrices with respect to general- ized equivalence, Visnyk Lviv. Univ., 61, 2003, pp. 153-160. [11] O.Helmer, The elementary divisor theorem for certain rings without chain condi- tion, Bull. Amer. Math. Soc., 49, 1943, pp. 225-236. [12] N.Ya.Komarnitskij, Commutative adequate Bezout domain and elementary divi- sor ring, Algebr. Issled., Collection of papers, Institute of mathematics NAN of Ukraine, Kiev, 24, 1996, pp. 97–113 (in Russian). [13] M.Ya.Komarnyts‘kij, V.M.Petrychkovych, Structure-theoretic properties of matri- ces over finitely generated principal ideal rings, Mat. Metody Fiz.-Mekh. Polya, 46, N. 2, 2003, pp. 7-21 (in Ukrainian). [14] V. M.Petrychkovych, On parallel factorizations of matrices over principal ideal rings, Mat. Metody Fiz.-Mekh. Polya, 40, N. 4, 1997, pp. 96-100 (in Ukrainian). Contact information Vasyl‘ M. Petrychkovych Department of Algebra, Pidstryhach Insti- tute for Applied Problems of Mechanics and the Mathematics National Academy of Sci- ences of Ukraine, 3B Naukova Str., Lviv, 79053, Ukraine E-Mail: vas_petrych@yahoo.com, vpetrych@iapmm.lviv.ua Received by the editors: 21.04.2004 and final form in 25.05.2004.

Generalized equivalence of collections of matrices and common divisors of matrices

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