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


Matrix of ℤ-module1

Yuichi Futa / Hiroyuki Okazaki / Yasunari Shidama
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


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


  • [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

Citation Information: Formalized Mathematics, Volume 23, Issue 1, Pages 29–49, 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

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Yuichi Futa and Yasunari Shidama
Formalized Mathematics, 2017, Volume 25, Number 2

Comments (0)

Please log in or register to comment.
Log in