Science Journal of Education
Volume 5, Issue 3, June 2017, Pages: 115-118
Received: Apr. 21, 2017;
Published: Apr. 21, 2017
Views 1688 Downloads 94
Jing Yao, Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
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.
Statistical Simulation for the Invertibility Test of Binary Random Matrices, Science Journal of Education.
Vol. 5, No. 3,
2017, pp. 115-118.
Sullivan, P.; Clarke D.; Clarke B. “Perspectives on mathematics, learning, and teaching”, Mathematics Teacher Education, 2013, pp. 7-12
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
Day, J. M.; Kalman, D. “Teaching linear algebra: issues and resources.” The College Mathematics Journal, 2001, 32(3), pp. 162-168
Ford W. Numerical Linear algebra with applications using MATLAB. 2015, pp. 79-101
Edelman, A. Eigenvalues and condition numbers of random matrices. MIT Dissertation, 1989, 106 pages
Strang, G. Essays in linear algebra. Wellesley-Cambridge Press, 2012, pp. iv-vi
Strang, G. Linear algebra and its applications. 4ed (international student edition), Brooks/Cole, Cengage Learning, 2006, pp. 65
Ryser, H. J. “Combinatorial properties of matrices of zeros and ones.” Canad. J. Math. 1957(9), pp. 371-377
Weisstein, E. W. “Weisstein's conjecture.” From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ WeissteinsConjecture.html
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
Rice, J. A. Mathematical Statistics and Data Analysis.3ed, Thomson, 2007, pp. 37-47