Matrix of ℤ-module1

Yuichi Futa; Hiroyuki Okazaki; Yasunari Shidama

Formalized Mathematics (2015)

  • Volume: 23, Issue: 1, page 29-49
  • ISSN: 1426-2630

Abstract

top
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.

How to cite

top

Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. "Matrix of ℤ-module1." Formalized Mathematics 23.1 (2015): 29-49. <http://eudml.org/doc/270886>.

@article{YuichiFuta2015,
abstract = {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.},
author = {Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama},
journal = {Formalized Mathematics},
keywords = {matrix of Z-module; matrix of linear transformation; bilinear form; matrix of -module},
language = {eng},
number = {1},
pages = {29-49},
title = {Matrix of ℤ-module1},
url = {http://eudml.org/doc/270886},
volume = {23},
year = {2015},
}

TY - JOUR
AU - Yuichi Futa
AU - Hiroyuki Okazaki
AU - Yasunari Shidama
TI - Matrix of ℤ-module1
JO - Formalized Mathematics
PY - 2015
VL - 23
IS - 1
SP - 29
EP - 49
AB - 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.
LA - eng
KW - matrix of Z-module; matrix of linear transformation; bilinear form; matrix of -module
UR - http://eudml.org/doc/270886
ER -

References

top
  1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990. 
  2. [2] Grzegorz Bancerek. Curried and uncurried functions. Formalized Mathematics, 1(3): 537–541, 1990. 
  3. [3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990. Zbl06213858
  4. [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990. 
  5. [5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990. 
  6. [6] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990. 
  7. [7] Czesław Byliński. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643–649, 1990. 
  8. [8] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990. 
  9. [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990. 
  10. [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990. 
  11. [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990. 
  12. [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990. 
  13. [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990. 
  14. [14] Wolfgang Ebeling. Lattices and Codes. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 2013. 
  15. [15] Yuichi Futa, Hiroyuki Okazaki, and Yasunari Shidama. ℤ-modules. Formalized Mathematics, 20(1):47–59, 2012. doi:10.2478/v10037-012-0007-z. Zbl1276.94012
  16. [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. Zbl06213848
  17. [17] Katarzyna Jankowska. Matrices. Abelian group of matrices. Formalized Mathematics, 2 (4):475–480, 1991. 
  18. [18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841–845, 1990. 
  19. [19] Jarosław Kotowicz. Bilinear functionals in vector spaces. Formalized Mathematics, 11(1): 69–86, 2003. 
  20. [20] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990. 
  21. [21] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335–342, 1990. 
  22. [22] Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: a cryptographic perspective. The International Series in Engineering and Computer Science, 2002. Zbl1140.94010
  23. [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. 
  24. [24] Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339–345, 1996. 
  25. [25] Michał Muzalewski. Rings and modules – part II. Formalized Mathematics, 2(4):579–585, 1991. 
  26. [26] Bogdan Nowak and Andrzej Trybulec. Hahn-Banach theorem. Formalized Mathematics, 4(1):29–34, 1993. 
  27. [27] Karol Pąk and Andrzej Trybulec. Laplace expansion. Formalized Mathematics, 15(3): 143–150, 2007. doi:10.2478/v10037-007-0016-5. 
  28. [28] Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29–34, 1999. 
  29. [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. 
  30. [30] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329–334, 1990. 
  31. [31] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990. 
  32. [32] Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569–573, 1990. 
  33. [33] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575–579, 1990. 
  34. [34] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990. 
  35. [35] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291–296, 1990. 
  36. [36] Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1(5):877–882, 1990. 
  37. [37] Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883–885, 1990. 
  38. [38] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990. 
  39. [39] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. 
  40. [40] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990. 
  41. [41] Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205–211, 1992. 
  42. [42] Katarzyna Zawadzka. The product and the determinant of matrices with entries in a field. Formalized Mathematics, 4(1):1–8, 1993. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.