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)].


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


Zbl 0247.15004
Full Text: DOI


[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
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