Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T11:51:20.151Z Has data issue: false hasContentIssue false
  • English
  • Français

Common solutions to a pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2

Published online by Cambridge University Press:  24 October 2008

Sujit Kumar Mitra
Sujit Kumar Mitra
Affiliation:
Indiana University, Bloomington
Get access Rights & Permissions [Opens in a new window]

Extract

Penrose (4) gave a necessary and sufficient condition for the consistency of the linear matrix equation AXB = C and also its complete class of solutions. A necessary and sufficient condition for the equations AX = C, XB = D to have a common solution was given by Cecioni (3) and an expression for the general common solution by Rao and Mitra ((6), p. 25). In the present paper, we obtain a necessary and sufficient condition for the equations A1XB1 = C1 and A2XB2 = C2 to have a common solution and also an expression for the general common solution. This result isuseful in computing a constrained inverse of a matrix, a concept originallyintroduced by Bott and Duffin(2) and recently extended by Rao and Mitra(7) who consider more general constraints with the object of bringing together the various generalized inverses and pseudoinverses under a common classification scheme.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 2 , September 1973 , pp. 213 - 216
Copyright
Copyright © Cambridge Philosophical Society 1973

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)Anderson, W. N. Jr and Duffin, R. J.Series and parallel addition of matrices. J. Math. Anal. Appl. 40 (1969), 576–594.CrossRefGoogle Scholar
(2)Borr, R. and Duffin, R. J.On the algebra of networks. Trans. Amer. Math. Soc. 74 (1953), 99–109.Google Scholar
(3)Cecioni, F.Sopra operazioni algebriche. Ann. Scuola. Norm. Sup. Pisa 11 (1910), 17–20.Google Scholar
(4)Penrose, R.A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51 (1955), 406–413.CrossRefGoogle Scholar
(5)Rao, C. R. Generalized inverse for matrices and its applications in mathematical statistics. Research papers in statistics. Festschrift for J. Neyman (Wiley, 1965).Google Scholar
(6)Rao, C. R. and Mitra, S. K.Generalized inverse of matrices and its applications (Wiley, 1971).Google Scholar
(7)Rao, C. R. and Mitra, S. K.Theory and application of constrained inverse of matrices. SIAM J. Appl. Math. 24 (1973), 473–488.CrossRefGoogle Scholar