Statistical Simulation for the Invertibility Test of Binary Random Matrices

Jing Yao

doi:doi:10.11648/j.sjedu.20170503.17

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Statistical Simulation for the Invertibility Test of Binary Random Matrices

Jing Yao

Published in Science Journal of Education (Volume 5, Issue 3)

Received: 21 April 2017 Accepted: Published: 21 April 2017

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Abstract

One specific mathematical problem is discussed by combining the knowledge of statistical simulation and linear algebra. Aiming to solve this easy-to-understand yet hard-to-answer problem, this paper tries in two ways to test the invertibility of large random binary matrices. By generating random entries of the matrices, and using sparse sampling strategies to get matrices, we also consider programming techniques in order to break the bottleneck of computing power. The proportion of singular matrices changes with the increase of matrix order and the trend is presented. The advantages and disadvantages of the methods are also analyzed from the aspects of result accuracy, time efficiency and applicability. This paper is an example of computer-aided teaching to assist students in enhancing their understanding and practical ability.

Published in	Science Journal of Education (Volume 5, Issue 3)
DOI	10.11648/j.sjedu.20170503.17
Page(s)	115-118
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Matrix Theory, Statistical Simulation, Sampling Strategy, College Mathematics Education

References

[1]	Sullivan, P.; Clarke D.; Clarke B. “Perspectives on mathematics, learning, and teaching”, Mathematics Teacher Education, 2013, pp. 7-12
[2]	Baturo, A.; Cooper, T; Doyle, K; Grant, E. “Using three levels in design of effective teacher-education tasks: The case of promoting conflicts with intuitive understandings in probability.” Journal of Mathematics Teacher Education, 2007, 10(4), pp. 251-259
[3]	Day, J. M.; Kalman, D. “Teaching linear algebra: issues and resources.” The College Mathematics Journal, 2001, 32(3), pp. 162-168
[4]	Ford W. Numerical Linear algebra with applications using MATLAB. 2015, pp. 79-101
[5]	Edelman, A. Eigenvalues and condition numbers of random matrices. MIT Dissertation, 1989, 106 pages
[6]	Strang, G. Essays in linear algebra. Wellesley-Cambridge Press, 2012, pp. iv-vi
[7]	Strang, G. Linear algebra and its applications. 4ed (international student edition), Brooks/Cole, Cengage Learning, 2006, pp. 65
[8]	Ryser, H. J. “Combinatorial properties of matrices of zeros and ones.” Canad. J. Math. 1957(9), pp. 371-377
[9]	Weisstein, E. W. “Weisstein's conjecture.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ WeissteinsConjecture.html
[10]	McKay, B. D.; Royle, G. F.; Wanless, I. M.; Oggier, F. E.; Sloane, N. J. A.; and Wilf, H. “Acyclic digraphs and eigenvalues of (0,1)-matrices.” J. Integer Sequences, 2004(7), Article 04.3.3, pp. 1-5
[11]	Rice, J. A. Mathematical Statistics and Data Analysis.3ed, Thomson, 2007, pp. 37-47

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Jing Yao. (2017). Statistical Simulation for the Invertibility Test of Binary Random Matrices. Science Journal of Education, 5(3), 115-118. https://doi.org/10.11648/j.sjedu.20170503.17

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Jing Yao. Statistical Simulation for the Invertibility Test of Binary Random Matrices. Sci. J. Educ. 2017, 5(3), 115-118. doi: 10.11648/j.sjedu.20170503.17

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AMA Style

Jing Yao. Statistical Simulation for the Invertibility Test of Binary Random Matrices. Sci J Educ. 2017;5(3):115-118. doi: 10.11648/j.sjedu.20170503.17

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@article{10.11648/j.sjedu.20170503.17,
  author = {Jing Yao},
  title = {Statistical Simulation for the Invertibility Test of Binary Random Matrices},
  journal = {Science Journal of Education},
  volume = {5},
  number = {3},
  pages = {115-118},
  doi = {10.11648/j.sjedu.20170503.17},
  url = {https://doi.org/10.11648/j.sjedu.20170503.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjedu.20170503.17},
  abstract = {One specific mathematical problem is discussed by combining the knowledge of statistical simulation and linear algebra. Aiming to solve this easy-to-understand yet hard-to-answer problem, this paper tries in two ways to test the invertibility of large random binary matrices. By generating random entries of the matrices, and using sparse sampling strategies to get matrices, we also consider programming techniques in order to break the bottleneck of computing power. The proportion of singular matrices changes with the increase of matrix order and the trend is presented. The advantages and disadvantages of the methods are also analyzed from the aspects of result accuracy, time efficiency and applicability. This paper is an example of computer-aided teaching to assist students in enhancing their understanding and practical ability.},
 year = {2017}
}

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TY  - JOUR
T1  - Statistical Simulation for the Invertibility Test of Binary Random Matrices
AU  - Jing Yao
Y1  - 2017/04/21
PY  - 2017
N1  - https://doi.org/10.11648/j.sjedu.20170503.17
DO  - 10.11648/j.sjedu.20170503.17
T2  - Science Journal of Education
JF  - Science Journal of Education
JO  - Science Journal of Education
SP  - 115
EP  - 118
PB  - Science Publishing Group
SN  - 2329-0897
UR  - https://doi.org/10.11648/j.sjedu.20170503.17
AB  - One specific mathematical problem is discussed by combining the knowledge of statistical simulation and linear algebra. Aiming to solve this easy-to-understand yet hard-to-answer problem, this paper tries in two ways to test the invertibility of large random binary matrices. By generating random entries of the matrices, and using sparse sampling strategies to get matrices, we also consider programming techniques in order to break the bottleneck of computing power. The proportion of singular matrices changes with the increase of matrix order and the trend is presented. The advantages and disadvantages of the methods are also analyzed from the aspects of result accuracy, time efficiency and applicability. This paper is an example of computer-aided teaching to assist students in enhancing their understanding and practical ability.
VL  - 5
IS  - 3
ER  -

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Author Information

Jing Yao

Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

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APA Style

Jing Yao. (2017). Statistical Simulation for the Invertibility Test of Binary Random Matrices. Science Journal of Education, 5(3), 115-118. https://doi.org/10.11648/j.sjedu.20170503.17

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ACS Style

Jing Yao. Statistical Simulation for the Invertibility Test of Binary Random Matrices. Sci. J. Educ. 2017, 5(3), 115-118. doi: 10.11648/j.sjedu.20170503.17

Copy | Download

AMA Style

Jing Yao. Statistical Simulation for the Invertibility Test of Binary Random Matrices. Sci J Educ. 2017;5(3):115-118. doi: 10.11648/j.sjedu.20170503.17

Copy | Download

@article{10.11648/j.sjedu.20170503.17,
  author = {Jing Yao},
  title = {Statistical Simulation for the Invertibility Test of Binary Random Matrices},
  journal = {Science Journal of Education},
  volume = {5},
  number = {3},
  pages = {115-118},
  doi = {10.11648/j.sjedu.20170503.17},
  url = {https://doi.org/10.11648/j.sjedu.20170503.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjedu.20170503.17},
  abstract = {One specific mathematical problem is discussed by combining the knowledge of statistical simulation and linear algebra. Aiming to solve this easy-to-understand yet hard-to-answer problem, this paper tries in two ways to test the invertibility of large random binary matrices. By generating random entries of the matrices, and using sparse sampling strategies to get matrices, we also consider programming techniques in order to break the bottleneck of computing power. The proportion of singular matrices changes with the increase of matrix order and the trend is presented. The advantages and disadvantages of the methods are also analyzed from the aspects of result accuracy, time efficiency and applicability. This paper is an example of computer-aided teaching to assist students in enhancing their understanding and practical ability.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Statistical Simulation for the Invertibility Test of Binary Random Matrices
AU  - Jing Yao
Y1  - 2017/04/21
PY  - 2017
N1  - https://doi.org/10.11648/j.sjedu.20170503.17
DO  - 10.11648/j.sjedu.20170503.17
T2  - Science Journal of Education
JF  - Science Journal of Education
JO  - Science Journal of Education
SP  - 115
EP  - 118
PB  - Science Publishing Group
SN  - 2329-0897
UR  - https://doi.org/10.11648/j.sjedu.20170503.17
AB  - One specific mathematical problem is discussed by combining the knowledge of statistical simulation and linear algebra. Aiming to solve this easy-to-understand yet hard-to-answer problem, this paper tries in two ways to test the invertibility of large random binary matrices. By generating random entries of the matrices, and using sparse sampling strategies to get matrices, we also consider programming techniques in order to break the bottleneck of computing power. The proportion of singular matrices changes with the increase of matrix order and the trend is presented. The advantages and disadvantages of the methods are also analyzed from the aspects of result accuracy, time efficiency and applicability. This paper is an example of computer-aided teaching to assist students in enhancing their understanding and practical ability.
VL  - 5
IS  - 3
ER  -

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