Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T07:32:36.003Z Has data issue: false hasContentIssue false
  • English
  • Français

Diagonal matrices

Published online by Cambridge University Press:  24 October 2008

H. W. Turnbull
H. W. Turnbull
Affiliation:
Trinity College, Professor of Mathematics, University of St Andrews
Get access Rights & Permissions [Opens in a new window]

Extract

In the following pages I have developed the theory of matrices by resolving them into parallel components arranged diagonally, rather than into the usual rows and columns. This treatment is natural in view of the fundamental fact that the resolution is undestroyed when matrices are formed into products (Theorem 2). It is closely related to the theory of continuants and of continued fractions. Certain features stand out in such a presentation—the distinction between the length and range of a diagonal (§ 4), that between regular and irregular diagonals (§ 6), and the use of equable partition (§ 7). The exact conditions for the existence of an rth root of a given singular matrix are examined in § 9 and summarized under the title, the condition of equability.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 29 , Issue 3 , 30 July 1933 , pp. 347 - 372
Copyright
Copyright © Cambridge Philosophical Society 1933

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

REFERENCES

(1)Sylvester, , Phil. Mag. (4) 5 (1853), 446; 6 (1853), 297; Math. Papers, 1, 609, 641.CrossRefGoogle Scholar
(2)Whittaker, E. T., Proc. Roy. Soc. Edinburgh, 36 (1915–1916), 243–255.CrossRefGoogle Scholar
(3)Turnbull, and Aitken, , Canonical Matrices (Glasgow, 1932).Google Scholar
(4)Rutherford, D. E., Proc. Edinburgh Math. Soc. (2) 3 (1932), 135–143.Google Scholar
Krishnamurthy Rao, S., “Invariant factors of a certain class of linear substitutions”, Jour. Indian Math. Soc. 19 (1932), 233–240.Google Scholar
(5)Frobenius, , Berliner Sitzungsberichte (1896), 7–16.Google Scholar
Baker, H. F., Proc. Cambridge Phil. Soc. 23 (1925), 22–27, who gives several references, dealing more particularly with the case of symmetric matrices.Google Scholar