Abstract
We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF x such that, for each λ in the domain ofF, F x (λ) is the inverse (resp. a generalized inverse) of the matrixF(λ). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BG] A. Ben-Israel and T. N. E. Greville,Generalized Inverses: Theory and Applications, John Wiley & Sons, New York London Sydney Toronto, 1974.
[BGK] H. Bart, I. Gohberg and M. A. Kaashoek,Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications, vol. 1 Birkhäuser Verlag, Basel-Boston-Stuttgart, 1979.
[BGKV] H. Bart, I. Gohberg, M. A. Kaashoek and P. Van Dooren, Factorization of Transfer Functions.SIAM J. Control and Optimization, vol. 18, no. 6, pp. 675–696, November 1980.
[BGR] J. A. Ball, I. Gohberg and L. Rodman, Realization and Interpolation of Rational Matrix Functions, inTopics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), Operator Theory: Advances and Applications, vol. 33, Birkhäuser Verlag, Basel Boston Berlin, pp. 1–72, 1988.
[BR1] J. A. Ball and M. Rakowski, Zero-Pole Structure of Nonregular Rational Matrix Functions, to appear.
[BR2] J. A. Ball and M. Rakowski, Null-Pole Subspaces of Nonregular Rational Matrix Functions, submitted for publication.
[C1] N. Cohen, On Minimal Factorization of Rational Matrix Functions,Integral Equations and Operator Theory, vol. 6, pp. 647–671, 1983.
[C2] N. Cohen, On Spectral Analysis and Factorization of Rational Matrix Functions, PhD Dissertaion, Weizmann Institute of Science, Rehovot, Israel August 1984.
[CG] K. Clancey and I. Gohberg,Factorization of Matrix Functions and Singular Integral Operators, Operator Theory: Advances and Applications, vol. 3, Birkhäuser Verlag, Basel-Boston-Stuttgart, 1981.
[CP] G. Conte and A. M. Perdon, On the Causal Factorization Problem,IEEE Transactions on Automatic Control, vol. AC, 30, no. 8, pp. 811–813, August 1985.
[CR] K. Clancey and M. Rakowski, Factorization of Rectangular Matrix Functions Relative to a Contour, unpublished note.
[DW] C. A. Desoer and B. H. Whalen, A Note on Pseudoinverses,J. Soc. Indust. Appl. Math., vol. 11, no. 2, pp. 442–447, June 1963.
[G] F. R. Gantmacher,The Theory of Matrices, vol. 2, Chelsea Publishing Company, New York, 1971.
[GK] I. Gohberg and M. A. Kaashoek, Regular Rational Matrix Functions with Prescribed Pole and Zero Structure, inTopics in Interpolation Theory of Rational Matrix Functions (ed. I. Gohberg), Operator Theory: Advances and Applications, vol. 33, Birkhäuser Verlag, Basel Boston Berlin, pp. 109–122, 1988.
[GKL] I. Gohberg, M. A. Kaashoek and P. Lancaster, Theory of Regular Matrix Polynomials and Band Toeplitz Operators,Integral Equations and Operator Theory, vol. 11, no. 6, pp. 776–882, 1988.
[K] J. K. Kang, Interpolation by Rational Matrix Functions with Minimal McMillan degree, PhD Dissertation, Blacksburg, Virginia, April 1990.
[M] C. C. MacDuffee,The Theory of Matrices, reprint by Chelsea Publishing Company, New York 1946.
[P] R. Penrose, A Generalized Inverse for Matrices,Proc. Cambridge Philos. Soc., vol.51, pp. 406–413, 1955.
[R] C. A. Rohde, Some Results on Generalized Inverses,SIAM Review, vol. 8, no. 2, pp. 201–205, April 1966.
[S] L. M. Silverman, Inversion of Multivariable Linear Systems,IEEE Transactions on Automatic Control, vol. AC-14, no. 3, pp. 270–276, June 1969.
[TL] A. E. Taylor and D. C. Lay,Introduction to Functional Analysis, Second Edition, John Wiley & Sons, New York Chichester Brisbane Toronto, 1980.
[V] P. Van Dooren, Factorization of a Rational Matrix: the Singular Case,Integral Equations and Operator Theory, vol. 7, pp. 704–741, 1984.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rakowski, M. Generalized pseudoinverses of matrix valued functions. Integr equ oper theory 14, 564–585 (1991). https://doi.org/10.1007/BF01204266
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01204266