Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year

SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
See all formats and pricing
More options …
Volume 23, Issue 1 (Mar 2015)


Matrix of ℤ-module1

Yuichi Futa
  • Japan Advanced Institute of Science and Technology, Ishikawa, Japan
/ Hiroyuki Okazaki
  • Shinshu University, Nagano, Japan
/ Yasunari Shidama
  • Shinshu University, Nagano, Japan
Published Online: 2015-03-31 | DOI: https://doi.org/10.2478/forma-2015-0003


In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22] and coding theory [14]. Some theorems in this article are described by translating theorems in [24], [26] and [19] into theorems of ℤ-module.

MSC: 11E39; 13C10; 03B35

Keywords: matrix of Z-module; matrix of linear transformation; bilinear form

MML: identifier: ZMATRLIN; version: 8.1.04 5.31.1231


  • [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.Google Scholar

  • [2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537–541, 1990.Google Scholar

  • [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Google Scholar

  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Google Scholar

  • [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Google Scholar

  • [6] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.Google Scholar

  • [7] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990.Google Scholar

  • [8] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.Google Scholar

  • [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Google Scholar

  • [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Google Scholar

  • [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.Google Scholar

  • [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Google Scholar

  • [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Google Scholar

  • [14] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013.Google Scholar

  • [15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi:10.2478/v10037-012-0007-z.CrossrefGoogle Scholar

  • [16] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. Free ℤ-module. Formalized Mathematics, 20(4):275–280, 2012. doi:10.2478/v10037-012-0033-x.CrossrefGoogle Scholar

  • [17] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2 (4):475–480, 1991.Google Scholar

  • [18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990.Google Scholar

  • [19] Jarosław Kotowicz. Bilinear functionals in vector spaces. Formalized Mathematics, 11(1): 69–86, 2003.Google Scholar

  • [20] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.Google Scholar

  • [21] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990.Google Scholar

  • [22] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002.Google Scholar

  • [23] Anna Justyna Milewska. The Hahn Banach theorem in the vector space over the field of complex numbers. Formalized Mathematics, 9(2):363–371, 2001.Google Scholar

  • [24] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339–345, 1996.Google Scholar

  • [25] Michał Muzalewski. Rings and modules – part II. Formalized Mathematics, 2(4):579–585, 1991.Google Scholar

  • [26] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993.Google Scholar

  • [27] Karol Pąk and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143–150, 2007. doi:10.2478/v10037-007-0016-5.CrossrefGoogle Scholar

  • [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999.Google Scholar

  • [29] Nobuyuki Tamura and Yatsuka Nakamura. Determinant and inverse of matrices of real elements. Formalized Mathematics, 15(3):127–136, 2007. doi:10.2478/v10037-007-0014-7.CrossrefGoogle Scholar

  • [30] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990.Google Scholar

  • [31] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Google Scholar

  • [32] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990.Google Scholar

  • [33] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990.Google Scholar

  • [34] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.Google Scholar

  • [35] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990.Google Scholar

  • [36] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877–882, 1990.Google Scholar

  • [37] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990.Google Scholar

  • [38] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Google Scholar

  • [39] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Google Scholar

  • [40] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.Google Scholar

  • [41] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205–211, 1992.Google Scholar

  • [42] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1–8, 1993.Google Scholar

About the article

Received: 2015-02-18

Published Online: 2015-03-31

Published in Print: 2015-03-01

1This work was supported by JSPS KAKENHI 21240001 and 22300285.

Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, DOI: https://doi.org/10.2478/forma-2015-0003.

Export Citation

© Yuichi Futa et al.. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. BY-SA 3.0

Comments (0)

Please log in or register to comment.
Log in