Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-15T07:12:31.734Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Differentials of Certain Matrix Functions

Published online by Cambridge University Press:  20 November 2018

David L. Powers
David L. Powers*
Affiliation:
Clarkson College of Technology, Potsdam, New York
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [5], Rinehart showed that if X is an n × n complex matrix with distinct eigenvalues, then a suitably defined diagonalizing matrix P and the diagonal matrix Λ of eigenvalues in P–lXP = Λ are both Hausdorff differentiate functions in an open set containing X. Furthermore, if the scalar function ƒ(z) is analytic at the eigenvalues of X, then the primary matrix function ƒ(X) is Hausdorff differentiable, and its differential may be represented in terms of the differentials of P and Λ [4]. Rinehart noted that the actual computation of differentials was difficult and ad hoc. This difficulty clearly arises because of the definition given for the diagonalizing matrix. Therefore, our aim in this note is to give a different definition of the diagonalizing matrix, one which simplifies the computations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 282 - 286
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Collar, A. R., Some notes on John's method for the improvement of approximate latent roots and vectors of a square matrix, Quart. J. Mech. Appl. Math. 1 (1948), 145–148.Google Scholar
2. Englefield, M. J., The commuting inverse of a square matrix, Proc. Cambridge Philos. Soc. 62 (1966), 667–671.Google Scholar
3. Neudecker, H., A note on Kronecker matrix products and matrix equation systems, SI AM J. Appl. Math. 17 (1969), 603–606.Google Scholar
4. Rinehart, R. F., The differential of a primary matrix function, Rend. Circ. Mat. Palermo 15 (1966), 209–215.Google Scholar
5. Rinehart, R. F., P and Din P-1XP = dg﹛λ1 …, λn﹜ = D as matrix functions of X, Can. J. Math. 18 (1966), 832–837.Google Scholar