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The Scientific World Journal
Volume 2014, Article ID 361349, 3 pages
http://dx.doi.org/10.1155/2014/361349
Research Article

A Note on the Drazin Indices of Square Matrices

1College of Automation, Harbin Engineering University, Harbin 150001, China
2College of Science, Harbin Engineering University, Harbin 150001, China
3College of Computer Science and Technology, Harbin Engineering University, Harbin 150001, China

Received 17 December 2013; Accepted 6 January 2014; Published 10 February 2014

Academic Editors: R. Alperin, V. Nitica, and M. Ronco

Copyright © 2014 Lijun Yu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For a square matrix , the smallest nonnegative integer such that rank () = rank () is called the Drazin index of . In this paper, we give some results on the Drazin indices of sum and product of square matrices.

1. Introduction

For , the smallest nonnegative integer such that is called the Drazin index of , denoted by . The Drazin inverse of is the unique matrix satisfying , and , where (see [13]). When , is called the group inverse of , denoted by . If , then . The theory of generalized inverses is an active research field in computational mathematics, and the Drazin index plays an important role in the study of the Drazin (group) inverse. Some results on the Drazin indices of matrices (operators) can be found in [49].

For any , it is known [10] that there exist nonsingular matrices and nilpotent matrix such that . In this case, , and is the smallest nonnegative integer such that ; that is, . Hence, is the smallest nonnegative integer such that the group inverse of exists.

In this paper, we give some results on the Drazin indices of sum and product of square matrices.

2. Some Lemmas

In order to prove our main results, we give some lemmas as follows.

Lemma 1 (see [7]). For any and nonnegative integer , the limit   exists if and only if .

Lemma 2 (see [8, 11]). Let be a square complex matrix, where is square. Then,

Lemma 3 (see [8, 11]). Let be a square complex matrix, where is square. Then,

Lemma 4 (see [12]). Let be a square complex matrix, where is square. Then, exists if and only if exists and .

Lemma 5 (see [12]). Let be a square complex matrix, where is square. Then, exists if and only if exist and .

Lemma 6 (see [9]). For any , .

3. Main Results

For a square matrix , let . We first give an upper bound for the Drazin index of the sum of two square matrices.

Theorem 7. Let be square complex matrices such that and . Then,

Proof. There exist nonsingular matrices and nilpotent matrix such that . Then, and . Suppose that , where is a square matrix with the same order as . By we get , . Then, . By we get .
Note that is the smallest nonnegative integer such that the group inverse of exists. Since , by Lemma 5, we have . Since , we have . Let . By Lemma 1, the limit exists. So . By Lemma 3, we get . Since , we have .

Now we give an example to show that the upper bound in Theorem 7 can be attained.

Example 8. Let , . Then, and . By computation, we have and . The Drazin index of is . So .
We can obtain the following result from Theorem 7.

Corollary 9. Let be square complex matrices such that . Then,

It is known that under the conditions , or , (see [13]). A better upper bound is given as follows.

Theorem 10. Let be a square complex matrix, where is square. Then, if one of the following holds:(1), ;(2), ;(3), ;(4), .

Proof. We only prove part (1). Parts (2)–(4) can be obtained in the same way.
Let , ; then, . By , , we get . By Corollary 9, we have By Lemma 2, we have

Theorem 11. For , if , then and are similar.

Proof. Clearly and are similar. So we only consider the case . There exist nonsingular matrices such that , where is the identity matrix of order . Suppose that , where . Then, Since , the group inverses of and both exist. Lemmas 4 and 5 imply that there exist matrices such that and . Then, Hence, and are similar.

Corollary 12. For , if , then and are similar for any .

Proof. By Lemma 6 and Theorem 11, and are similar for any .

The following corollary is a special case of Theorem 11.

Corollary 13. For , if and both exist, then and are similar.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant no. 11371109 and the Fundamental Research Funds for the Central Universities: HEUCF041417.

References

  1. C. Bu, L. Sun, J. Zhou, and Y. Wei, “Some results on the Drazin inverse of anti-triangular matrices,” Linear and Mutilinear Algebra, vol. 61, pp. 1568–1576, 2013. View at Google Scholar
  2. C. Bu and C. Zhang, “A note on the formulas for the Drazin inverse of the sum of two matrices,” Linear Algebra and Its Applications, vol. 439, pp. 565–576, 2013. View at Google Scholar
  3. C. Deng and Y. Wei, “Representations for the Drazin inverse of 2 × 2 block-operator matrix with singular Schur complement,” Linear Algebra and Its Applications, vol. 435, no. 11, pp. 2766–2783, 2011. View at Publisher · View at Google Scholar · View at Scopus
  4. R. Bru, J. J. Climent, and M. Neumann, “On the index of block upper triangular matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 436–447, 1995. View at Google Scholar
  5. M. Catral, D. D. Olesky, and P. van den Driessche, “Block representations of the Drazin inverse of a bipartite matrix,” Electronic Journal of Linear Algebra, vol. 18, pp. 98–107, 2009. View at Google Scholar · View at Scopus
  6. M. Catral, D. D. Olesky, and P. van den Driessche, “Block representations of the Drazin inverse of a bipartite matrix,” Electronic Journal of Linear Algebra, vol. 18, pp. 98–107, 2009. View at Google Scholar · View at Scopus
  7. C. D. Meyer Jr., “Limits and the index of a square matrix,” SIAM Journal on Applied Mathematics, vol. 26, no. 3, pp. 469–478, 1974. View at Google Scholar · View at Scopus
  8. C. D. Meyer and N. J. Rose, “The index and the Drazin inverse of block triangular matrices,” SIAM Journal on Matrix Analysis and Applications, vol. 33, pp. 1–7, 1977. View at Google Scholar
  9. Q. Xu, Y. Wei, and C. Song, “Explicit characterization of the Drazin index,” Linear Algebra and Its Applications, vol. 436, no. 7, pp. 2273–2298, 2012. View at Publisher · View at Google Scholar · View at Scopus
  10. A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, New York, NY, USA, 2nd edition, 2003.
  11. R. E. Hartwig and J. M. Shoaf, “Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices,” Journal of the Australian Mathematical Society A, vol. 24, pp. 10–34, 1977. View at Google Scholar
  12. C. Cao, “Some results of group inverses for partitioned matrices over skew fields,” Journal of Natural Science of Heilongjiang University, vol. 18, no. 3, pp. 5–7, 2001. View at Google Scholar
  13. D. S. Cvetković-Ilić, J. Chen, and Z. Xu, “Explicit representations of the Drazin inverse of block matrix and modified matrix,” Linear and Multilinear Algebra, vol. 57, no. 4, pp. 355–364, 2009. View at Publisher · View at Google Scholar · View at Scopus