×

An alternative limit expression of Drazin inverse and its application. (English) Zbl 0805.65041

Let \(A\) be an \(n\times n\) matrix and \(\text{ind}(A)\) the index of \(A\), the smallest nonnegative integer for which \(\text{rank}(A^ k)=\text{rank}(A^{k+1})\). The author presents a new limit expression for the Drazin inverse
(1) \(A^ D=\lim_{\lambda\to 0} (\lambda+A)^{-(\ell+1)} A^ \ell\), where \(\ell\geq k=\text{ind}(A)\). It is assumed that \(- \lambda\not\in\sigma(A)\), the set of all eigenvalues of \(A\). The limit expression (1) provides a new proof of a finite algorithm for the Drazin inverse given by T. N. E. Greville [Linear Algebra Appl. 6, 205-208 (1973; Zbl 0247.15004)].

MSC:

65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A09 Theory of matrix inversion and generalized inverses

Citations:

Zbl 0247.15004
Full Text: DOI

References:

[1] Barnett, S., Leverrier’s algorithm: A new proof and extensions, SIAM J. Matrix Anal. Appl., 10, 551-556 (1989) · Zbl 0682.65022
[2] Campbell, S. L.; Meyer, C. D., Generalized Inverses of Linear Transformations (1979), Thomson Press (India) Ltd: Thomson Press (India) Ltd New Delhi · Zbl 0417.15002
[3] Decell, H. P., An application of the Cayley-Hamilton theorem to generalized matrix inversion, SIAM Rev., 7, 526-528 (1965) · Zbl 0178.35504
[4] Drazin, M. P., Pseudo-inverses in associative rings and semigroups, Amer. Math. Monthly, 65, 506-514 (1958) · Zbl 0083.02901
[5] Gower, J. C., A modified Leverrier-Faddeev algorithm for matrices with multiple eigenvalues, Linear Algebra Appl., 31, 61-70 (1980) · Zbl 0435.65028
[6] Greville, T. N.E., The Souriau-Frame algorithm and the Drazin pseudoinverse, Linear Algebra Appl., 6, 205-208 (1973) · Zbl 0247.15004
[7] Hartwig, R. E., More on the Souriau-Frame algorithm and the Drazin inverse, SIAM J. Appl. Math., 31, 42-46 (1976) · Zbl 0335.15008
[8] Ji, J., The algebraic pertubation method for generalized inverses, J. Comput. Math., 7, 327-333 (1989) · Zbl 0716.15004
[9] Kalaba, R. E.; Wang, J. S., A new proof for Decell’s finite algorithm for the generalized inverse, Appl. Math. Comput., 12, 199-211 (1983) · Zbl 0546.65015
[10] Lewis, F. L., Further remarks on the Cayley-Hamilton theorem and Leverrier’s method for the matrix pencil (sE - A), IEEE Trans. Automat. Control, 31, 869-870 (1986) · Zbl 0601.15010
[11] Wang, G. R., A finite algorithm for computing the weighted Moore-Penrose inverse \(A^+_{ MN } \), Appl. Math. Comput., 23, 277-289 (1987) · Zbl 0635.65039
[12] Wang, G. R.; Lin, Y., A new extension of Leverrier’s algorithm, Linear Algebra Appl., 180, 227-238 (1993) · Zbl 0782.65062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.