Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T09:04:08.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Eigencircles and associated surfaces

Published online by Cambridge University Press:  23 January 2015

M. J. Englefield  and
G.E. Farr
M. J. Englefield
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, Victoria 3800, Australia
G.E. Farr
Affiliation:
Clayton School of Information Technology, Monash University, Clayton, Victoria 3800, Australia, e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

Linear algebra has many fruitful connections with geometry. This article develops one such connection: the relationship between a 2 × 2 matrix and an associated circle which we call the eigencircle.

This connection was first investigated in a previous paper of ours [1], but the present paper is self-contained, and in fact introduces eigencircles in a different way. Here we discuss some surfaces containing the eigencircle which also have a number of interesting properties and connections with the associated matrix.

Type
Articles
Information
The Mathematical Gazette , Volume 94 , Issue 531 , November 2010 , pp. 438 - 449
Copyright
Copyright © The Mathematical Association 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Englefield, M. J. and Farr, G. E., Eigencircles of 2 × 2 matrices, Mathematics Magazine 79 (October 2006) pp. 281–289.10.1080/0025570X.2006.11953417Google Scholar
2. Carmichael, R. D., Boundary value and expansion problems: algebraic basis of the theory, Amer. J. Math. 43 (1921) pp. 69–101.10.2307/2370243Google Scholar
3. Atkinson, F. V., Multiparameter spectral theory, Bull. Amer. Math. Soc. 74 (1968) pp. 1–27.10.1090/S0002-9904-1968-11866-XGoogle Scholar
4. Binding, P. and Browne, P. J., Two parameter eigenvalue problems for matrices, Linear Algebra Appl. 113 (1989) pp. 139–157.10.1016/0024-3795(89)90293-0Google Scholar
5. Sleeman, B. D., Multiparameter spectral theory in Hilbert Space, Research Notes in Mathematics 22, Pitman, London (1978).10.1016/0022-247X(78)90160-9Google Scholar
6. Volkmer, H., Multiparameter eigenvalue problems and expansion theorems, Lecture notes in mathematics 1356, Springer-Verlag, Berlin (1988).10.1007/BFb0089295Google Scholar
7. Heath, T. L., The thirteen books of the Elements, Vol. 2: Books III-IX (2nd edn.), Dover, New York (1956).Google Scholar
8. Roe, J., Elementary geometry, Oxford University Press (1993).Google Scholar
9. Prasolov, V. V., Problems and theorems in linear algebra, American Mathematical Society (1994).10.1090/mmono/134Google Scholar
10. Farr, G. E., An eigencircle applet, http://www.csse.monash.edu.au/~gfarr/researchleigencircles.html Google Scholar