Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T20:24:53.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix Inversion by Partitioning

Published online by Cambridge University Press:  07 June 2016

Eryk Kosko
Eryk Kosko*
Affiliation:
Avro Aircraft Ltd., Toronto
Get access

Summary

A systematic discussion of partitioning as a tool for matrix inversion is presented, together with various methods and applications which have been of help in actual computations. New concepts are introduced, among them those of super-matrix and of square partitioning. The most usual type of partitioning, that into 2x2 sub-matrices, is discussed in detail, showing the orderly arrangement of the calculations in an auxiliary matrix. Further sections deal with matrices of the continuant type, and with special types of symmetry in the arrangement of the sub-matrices. The greatest advantage of the method of partitioning for the inversion of these types (as compared with the elimination method) lies in a considerable reduction in the number of arithmetical operations.

Type
Research Article
Information
Aeronautical Quarterly , Volume 8 , Issue 2 , May 1957 , pp. 157 - 184
Copyright
Copyright © Royal Aeronautical Society. 1957

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. Aitken, A. C. Determinants and Matrices. Oliver and Boyd, Edinburgh and London, 1938; also Fourth Edition, Interscience Publishers, New York, 1946.Google Scholar
2. Frazer, R. A., Duncan, W. J. and Collar, A. R. Elementary Matrices. Cambridge University Press, 1935; also reprint, Macmillan, New York, 1947.Google Scholar
3. Michal, A. D. Matrix and Tensor Calculus, with Applications to Mechanics, Elasticity and Aeronautics. John Wiley, New York, 1947.Google Scholar
4. Perlis, S. Theory of Matrices. Addison-Wesley Press, Cambridge, Massachusetts, 1952.Google Scholar
5. Wade, Th.L. The Algebra of Vectors and Matrices. Addison-Wesley Press, Cambridge, Massachusetts, 1951.Google Scholar
6. Zurmuhl, R. Matrizen. J. Springer, Berlin-Göttingen-Heidelberg, 1950.Google Scholar
7. Duncan, W. J. Some Devices for the Solution of Large Sets of Simultaneous Equations (with an Appendix on the Reciprocation of Partitioned Matrices). Phil, Mag., Series 7, Vol. 35, p. 660, 1944.CrossRefGoogle Scholar
8. Hotelling, H. Some New Methods in Matrix Calculation. Annals of Mathematical Statistics, Vol. 14, 1943.CrossRefGoogle Scholar
9. Morrison, I. F. The Solution of Three-Term Simultaneous Linear Equations by the Use of Submatrices. The Engineering Journal (Canada), Vol. 29, 1946.Google Scholar
10. Berger, W. I. and Saibel, E. On the Inversion of Continuant Matrices. Journal of the Franklin Institute, Vol. 256, p. 249, 1953.Google Scholar
11. Kosko, E. Effect of Local Modifications in Redundant Structures. Journal of the Aeronautical Sciences, Vol. 21, p. 206, 1954.Google Scholar
12. Brock, John E. Variations of Coefficients of Simultaneous Linear equations. quarterly of Applied Mathematics, Vol 11, p. 234, 1953.Google Scholar
13. Von Neumann, J. and Goldstine, H. H. Numerical Inverting of Matrices of High Order. Bulletin of the American Mathematical Society, Vol. 53, 1947.CrossRefGoogle Scholar
14. Duncan, W. J. Reciprocation of Triply Partitioned Matrices. Journal of the Royal Aeronautical Society, Vol. 60, p. 131, February 1956.CrossRefGoogle Scholar
15. Kosko, E. Reciprocation of Triply Partitioned Matrices. Journal of the Royal Aeronautical Society, Vol. 60, p. 490, July 1956.CrossRefGoogle Scholar