On the Singular Value Decomposition and Ranking Techniques
Zizler Peter 1, Thangarajah Pamini 2, Sobhanzadeh Mandana 3
Mount Royal University
4825 Mt Royal Gate SW
Calgary, AB T3E 6K6
1 E-mail: pzizler@mtroyal.ca
2 E-mail: pthangarajah@mtroyal.ca
3 E-mail: msobhanzadeh@mtroyal.ca
Received:
Received: 13 November 2019; revised: 18 March 2020; accepted: 18 March 2020; published online: 21 March 2020
DOI: 10.12921/cmst.2019.0000048
Abstract:
Let A be a positive non-singular n×n matrix. An approximation for a positive eigenvector for A*A corresponding to the dominant singular value of A was suggested as the normalized version of a weighted sum of the rows of A with weights being the euclidean norms of the rows of A. In our paper we give a justification for this approach via the iteration of the power method and we show numerically that choosing the l1 norm yields better results. Applications of our results are given to ranking techniques.
Key words:
References:
[1] G. Benettin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application, Meccanica 15, 21–30 (1980).
[2] C. Hepler, P. Thangarajah, P. Zizler, Ranking in Professional Sports: An Application of Linear Algebra for Computer Science Students, 21st Western Canadian Conference on Computing Education, Kamloops, BC, Canada (2016).
[3] R.A. Hanneman, M. Riddle, Concepts and Measures for Basic Network Analysis, The Sage Handbook of Social Network Analysis, SAGE, 346–347 (2011).
[4] D. James, C. Botteron, Understanding Singular Vectors, The College Mathematics Journal 44(3), 220–226 (2013).
[5] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press (1985).
[6] S. Montesinos, P. Zizler, V. Zizler, An Introduction to Modern Analysis, Springer (2015).
[7] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems, Progress of Theoretical Physics 61(6), 1605–1616 (1979).
[8] G. Strang, Linear Algebra and Its Applications, Cengage (previously Brooks/Cole), 4th edition (2006).
Let A be a positive non-singular n×n matrix. An approximation for a positive eigenvector for A*A corresponding to the dominant singular value of A was suggested as the normalized version of a weighted sum of the rows of A with weights being the euclidean norms of the rows of A. In our paper we give a justification for this approach via the iteration of the power method and we show numerically that choosing the l1 norm yields better results. Applications of our results are given to ranking techniques.
Key words:
References:
[1] G. Benettin, L. Galgani, A. Giorgilli, J.M. Strelcyn, Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application, Meccanica 15, 21–30 (1980).
[2] C. Hepler, P. Thangarajah, P. Zizler, Ranking in Professional Sports: An Application of Linear Algebra for Computer Science Students, 21st Western Canadian Conference on Computing Education, Kamloops, BC, Canada (2016).
[3] R.A. Hanneman, M. Riddle, Concepts and Measures for Basic Network Analysis, The Sage Handbook of Social Network Analysis, SAGE, 346–347 (2011).
[4] D. James, C. Botteron, Understanding Singular Vectors, The College Mathematics Journal 44(3), 220–226 (2013).
[5] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press (1985).
[6] S. Montesinos, P. Zizler, V. Zizler, An Introduction to Modern Analysis, Springer (2015).
[7] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems, Progress of Theoretical Physics 61(6), 1605–1616 (1979).
[8] G. Strang, Linear Algebra and Its Applications, Cengage (previously Brooks/Cole), 4th edition (2006).
