The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem

Liu, Na; Wang, Hongxing

doi:https://doi.org/10.1155/2021/4952943

Journal of Mathematics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Fuzzy Approximation and its Applications in the Stabilization of Dynamic Systems 2021

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 4952943 | https://doi.org/10.1155/2021/4952943

The Characterizations of WG Matrix and Its Generalized Cayley–Hamilton Theorem

Na Liu¹and Hongxing Wang¹

Academic Editor: Efthymios G. Tsionas

Received22 Oct 2021

Accepted27 Nov 2021

Published27 Dec 2021

Abstract

Based on the core-EP decomposition, we use the WG inverse, Drazin inverse, and other inverses to give some new characterizations of the WG matrix. Furthermore, we generalize the Cayley–Hamilton theorem for special matrices including the WG matrix. Finally, we give examples to verify these results.

1. Introduction

First, we use the following notations. Let stand for the set of complex matrices. The symbols , , , and represent the conjugate transpose, range, rank, and determinant of , respectively. The smallest positive integer such that is called the index of and it is denoted by . The Moore–Penrose inverse of is the unique matrix satisfying the following equations:and the unique matrix is denoted by [1, 2]. Furthermore, we denote

The Drazin inverse of is the unique matrix such that

and the unique matrix is usually denoted by , where [1, 2]. In particular, when , is called the group inverse of and is denoted by . Therefore, we call it a group invertible matrix with index 1. The symbol stands for the set of group invertible matrices in :

Baksalary and Trenkler [3] defined the core inverse of a complex matrix with index 1. Let ; the core inverse of is the unique matrix which satisfies the following equations:and it is denoted by . Subsequently, a variety of new generalized inverses have been established successively. Let with . The core-EP inverse of is the unique matrix satisfying , , and , and the core-EP inverse of is denoted by [4]; the B-T inverse of is the unique matrix satisfying , and the B-T inverse of is denoted by [5]; the DMP inverse of is the unique matrix satisfying , , and , and the DMP inverse of is denoted by [6]; the dual DMP inverse of is the unique matrix satisfying , , and , and the dual DMP inverse of is denoted by [6]; the CMP inverse of is the unique matrix satisfying , , and , and the CMP inverse of is denoted by [7]. It is easy to see that core-EP inverse and DMP inverse are both generalized core inverses, which are extensions of core inverse on square matrices without index constraint, and when , .

Furthermore, Wang and Chen [8] proposed a generalized group inverse. Let , if satisfies the following equations:where is called the WG inverse of , and is unique. It is usually denoted by . By applying the definition, we can obtain and . It is noteworthy that and , when . Then, Ferryra et al. [9] extended the definition of WG inverse to the general matrix, defined the weighted WG inverse, and gave its expression, properties, and characterizations; Mosić and Zhang [10] established the weighted WG inverse of Hilbert space operator; Zhou et al. [11] generalized WG inverse to a proper -ring and gave a new characterization of WG inverse; Zhou et al. [12] generalized m-WG inverse to a unitary ring with involution and gave some properties of m-WG inverse; Mosić and Stanimirović [13] gave new characterizations, limit representations, integral representations, and perturbation formulae of the WG inverse.

By applying the WG inverse, Wang and Liu [14] introduced the definition of WG matrix based on the properties and characterizations of WG inverse. Let ; if commutes with its WG inverse, is called WG matrix. The symbol stands for the set of WG matrices in [14]:

Subsequently, Yan et al. [15] investigated some new characterizations of weak group inverse by projection, the Bott–Duffin inverse, etc.

Matrix decomposition is very important, which not only functions as a significant role in every branch of mathematics but also has a wide range of applications in engineering. With the development of new generalized inverses, new research tools such as matrix decomposition and algorithm are also given. Wang established the core-EP decomposition of square matrix over complex fields [16]. Core-EP decomposition is one of the commonly used tools to study core-EP inverse and several new generalized inverses.

Lemma 1 (see [16], core-EP decomposition). Let with . Then, there exist and , such that , where , , and . Furthermore,where is a unitary matrix, is nonsingular, and is nilpotent.
By applying the above decomposition, it is easy to verify

Lemma 2 (see [14]). Let with , then and

Lemma 3 (see [8, 14]). Let be as in the form (8). Among , , , and , any two of them are equivalent. If , thenwhere is a positive integer and is the index of .

Lemma 4 (see [16–20]). Let be as in the form [21], then(1)(2)(3)(4)(5)(6)(7)where , , , and .

The classical Cayley–Hamilton theorem is one of the most important theorems in matrix theory. On the basis of the classical Cayley–Hamilton theorem, mathematicians established the rectangular matrix, block matrix, pair of block matrix, and other matrices as well as more generalized Cayley–Hamilton theorem for generalized inverse matrices. They also gave application of generalized Cayley–Hamilton theorem in several control systems [22–25]. In [26], Wang, Chen, and Yan gave the generalized Cayley–Hamilton theorem of core-EP inverse matrix and DMP inverse matrix by core-EP decomposition, and the characteristic polynomial equations of core-EP inverse matrix and DMP inverse matrix were also discussed. In [2, 27], the researchers studied the applications of generalized Cayley–Hamilton theorems in generalized inverses such as Drazin inverse and Moore–Penrose inverse. Based on the above researches, this paper will focus on the WG matrix, the equivalent characterizations of WG matrix, and the generalized Cayley–Hamilton theorem for special matrices including WG matrix.

2. Some Characterizations of WG Matrix

In [14], the definition and characterizations of WG matrix are given through the commutativity of matrix and WG inverse. In [15], Yan et al. investigated some new characterizations of WG matrix.

Theorem 1 (see [15]). Let with , then the following conditions are equivalent:(1);(2);(3);(4);(5);(6);(7).

It is pointed out that the set of group invertible matrices is a subset of WG matrices set. Special matrices such as WG matrix, group matrix, EP, -EP, and -EP matrix have rich intersection [14]. In this section, we will mainly apply core-EP decomposition to study the characterization of WG matrix.

Theorem 2. Let with , then the following conditions are equivalent:(1), where and are as in the form [21];(2);(3);(4);(5) commutes with ;(6) is a WG matrix for any unitary matrix ;(7) commutes with ;(8) commutes with ;(9) = ;(10) commutes with ;(11) = ;(12) = ;(13);(14);(15);(16);(17);(18);(19);(20);(21);(22) commutes with ;(23);(24);(25);(26);(27);(28);(29) commutes with ;(30);(31);(32);(33);(34);(35);(36);

Proof. From [14], we know that Conditions (1)–(5) are equivalent.
Let be a unitary matrix, then andThus, Conditions (2) and (6) are equivalent.
Let the core-EP decomposition of be as in the form [21]. By using Lemma 3, we obtainApplying [21, 27], we haveBy comparing [4, 7], we can getThus, Conditions (1) and (7) are equivalent.
By Lemma 4, we obtainBy applying [10, 13], we haveFrom what has been discussed above, we can surely come to the conclusion thatThus, Conditions (1) and (8) are equivalent.
Because of [7, 16], we can getThus, Conditions (1) and (9) are equivalent.
From [6, 23], we obtainBy comparing the above equations, we haveThus, Conditions (1) and (10) are equivalent.
By using [19], we can getBy applying the above equations, we obtainThus, Conditions (1) and (11) are equivalent.
Applying [10, 13], we haveBy comparing the above equations, we can getThus, Conditions (1) and (12) are equivalent.
By [19], we obtainFrom the above equation and [6], we can getThus, Conditions (1) and (13) are equivalent.
By applying [13, 23], we haveBy comparing the above equation and [6], we can getThus, Conditions (1) and (14) are equivalent.
Because of [6, 27], we obtainBy the above equation and [27], we haveThus, Conditions (1) and (15) are equivalent.
From [6, 27], we can getBy comparing the above equation and [6], we obtainHence, Conditions (1) and (16) are equivalent.
By using Lemma 2, we haveFrom the above equation and [23], we obtainThus, Conditions (1) and (17) are equivalent.
By applying Lemma 4, we can getBy (41), we obtainBy applying the above formula and (39), we have if and only if . Thus, Conditions (1) and (18) are equivalent.
By applying Lemma 4, we can getBecause of (43), we obtainBy comparing the above formula and (39), we obtain if and only if . Thus, Conditions (1) and (19) are equivalent.
Applying [6, 27], we haveIf , then . Conversely, let ; we get , that is, .
If the index of is equal to 1, then , that is, . If the index of is equal to 2, then . From , we have . Since is invertible, we obtain . Let the index of be more than or equal to 3, then and . If postmultiplication by , then . Since is invertible and , then . Furthermore, postmultiplication by , then . Since is invertible and , then . By repeating the process times, we have .
From what has been discussed above, if and only if . Thus, Conditions (1) and (20) are equivalent.
By [6, 21, 23], we obtainBy comparing the above equations, we can getThus, Conditions (1) and (21) are equivalent.
By applying (35) and (37), we have if and only if . Thus, Conditions (1) and (22) are equivalent.
Because of [19], we haveBy comparing the above equation and [7], we obtain . is equivalent to . Hence, Conditions (1) and (23) are equivalent.
From [2, 23], we have if and only if . Thus, Conditions (1) and (24) are equivalent.
By using Lemma 4, we obtainBy using (49), we haveBy comparing the above formula and [2], we can get if and only if . Thus, Conditions (1) and (25) are equivalent.
By Lemma 4, we obtainBy applying (51), we haveApplying the above formula and [2], we can get if and only if . Hence, Conditions (1) and (26) are equivalent.
Because of Lemma 4, we obtainFrom (53), we haveBy comparing the above formula and [2], we can get if and only if . Hence, Conditions (1) and (27) are equivalent.
By using Lemma 1 and Lemma 4, we obtainBy using the above formula, we haveIf , we can easily prove that . Conversely, if the index of is equal to 1 or 2, then obviously . Let the index of be more than or equal to 3; we get , that is, . Postmultiplying the above equation by , then , that is, . Through , we have . In the same way, we obtain , that is, . From what has been discussed above, we have if and only if . Hence, Conditions (1) and (28) are equivalent.
By Lemma 1 and Lemma 4, we can getBy applying [23] and (57), we haveSince the above formula, and are equivalent to . Thus, Conditions (1) and (29) are equivalent.
Because of [13] and (57), we can getIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (30) are equivalent.
From (59), we haveIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (31) are equivalent.
By using [13], (43), and (59), we can getIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Thus, Conditions (1) and (32) are equivalent.
Applying Lemma 4, [10], and (59), we obtainIf , then . Conversely, let , then and . By using the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (33) are equivalent.
By Lemma 4, [13], and (59), we haveIf , then . Conversely, let , then and . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (34) are equivalent.
By applying [4, 8], we obtain if and only if . Hence, Conditions (1) and (35) are equivalent.
Because of [16], (57) andIf , then . Conversely, let , then , that is, . By applying the method which is used to verify the equivalence of Conditions (1) and (28), we obtain . From what has been discussed above, is equivalent to . Hence, Conditions (1) and (36) are equivalent.

3. Generalized Cayley–Hamilton Theorem

In this section, we extend the classical Cayley–Hamilton theorem to some special matrix such as the WG matrix.

Theorem 3 (see [21]). Let ; the characteristic polynomial of isthenIn [21], if is singular, then .

Theorem 4. Let be singular with . Ifthenwhere is the weak group inverse of the matrix .

Proof. Let be singular; we use the Cayley–Hamilton theorem, thenPostmultiplying the above equation by , we getBy using the properties of the WG matrix, that is, , we have . A similar method can be used to obtain , , , and . Substituting the above equations into (71), we have (69).

Example 1. LetIt is easy to confirm that the weak group inverse isThen,From the classical Cayley–Hamilton theorem, we haveBy applying Theorem 4, we obtainNext, we extend the classical Cayley–Hamilton theorem to the WG inverse matrix. Let with . From Lemma 1 and Lemma 2, it can be obtainedThe characteristic polynomial of isIt is given by the classical Cayley–Hamilton theoremPostmultiplying the above equation by ,By using (77) and (78), we have Theorem 5.

Theorem 5. Let with , then the characteristic polynomial of iswhere is as in (78).

Example 2. Letthen , andFurthermore,

4. Conclusion

In this paper, we give some new characterizations of the WG matrix. Some relations between various well-known generalized inverses and the WG matrix are proved, and generalized Cayley–Hamilton theorem for special matrices such as WG matrix is given. We are convinced that researches about the WG matrix will also gain more attention in the near future. Some perspectives for further researches can be described as follows: (1) our further goal will be to investigate the definition, properties, and characterizations of tensor WG matrix; (2) further, we will study the iterative algorithm of the tensor WG matrix and its application [28].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported partially by the Innovation Project of Guangxi Graduate Education (grant no. YCSW2021146), the Special Fund for Bagui Scholars of Guangxi (grant no. 2016A17), Guangxi Natural Science Foundation (grant no. 2018GXNSFAA138181), and the Special Fund for Science and Technological Bases and Talents of Guangxi (grant no. GUIKE AD19245148).

References

S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2009.
G. R. Wang, Y. M. Wei, and S. Z. Qiao, Generalized Inverses: Theory and Computations, Springer, Singapore, Singapore, 2018.
O. M. Baksalary and G. Trenkler, “Core inverse of matrices,” Linear and Multilinear Algebra, vol. 58, no. 6, pp. 681–697, 2010.
View at: Publisher Site | Google Scholar
K. Manjunatha Prasad and K. S. Mohana, “Core-EP inverse,” Linear and Multilinear Algebra, vol. 62, no. 6, pp. 792–802, 2014.
View at: Publisher Site | Google Scholar
O. M. Baksalary and G. Trenkler, “On a generalized core inverse,” Applied Mathematics and Computation, vol. 236, pp. 450–457, 2014.
View at: Publisher Site | Google Scholar
S. B. Malik and N. Thome, “On a new generalized inverse for matrices of an arbitrary index,” Applied Mathematics and Computation, vol. 226, pp. 575–580, 2014.
View at: Publisher Site | Google Scholar
M. Mehdipour and A. Salemi, “On a new generalized inverse of matrices,” Linear and Multilinear Algebra, vol. 66, no. 5, pp. 1046–1053, 2018.
View at: Publisher Site | Google Scholar
H. Wang and J. Chen, “Weak group inverse,” Open Mathematics, vol. 16, no. 1, pp. 1218–1232, 2018.
View at: Publisher Site | Google Scholar
D. E. Ferreyra, V. Orquera, and N. Thome, “A weak group inverse for rectangular matrices,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 113, no. 4, pp. 3727–3740, 2019.
View at: Publisher Site | Google Scholar
D. Mosić and D. C. Zhang, “Weighted weak group inverse for Hilbert space operators,” Frontiers of Mathematics in China, vol. 15, no. 4, pp. 709–726, 2020.
View at: Publisher Site | Google Scholar
M. Zhou, J. Chen, and Y. Zhou, “Weak group inverses in proper ∗-rings,” Journal of Algebra and Its Applications, vol. 19, no. 12, Article ID 2050238, 2019.
View at: Publisher Site | Google Scholar
Y. Zhou, J. Chen, and M. Zhou, “m-weak group inverses in a ring with involution,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, vol. 115, no. 1, p. 2, 2020.
View at: Publisher Site | Google Scholar
D. Mosić and P. S. Stanimirovi, “Representations for the weak group inverse,” Applied Mathematics and Computation, vol. 397, no. 6, Article ID 125957, 2021.
View at: Publisher Site | Google Scholar
H. Wang and X. Liu, “The weak group matrix,” Aequationes Mathematicae, vol. 93, no. 6, pp. 1261–1273, 2019.
View at: Publisher Site | Google Scholar
H. Yan, H. X. Wang, H. Wang, K. Zuo, and Y. Chen, “Further characterizations of the weak group inverse of matrices and the weak group matrix,” AIMS Mathematics, vol. 6, no. 9, pp. 9322–9341, 2021.
View at: Publisher Site | Google Scholar
H. Wang, “Core-EP decomposition and its applications,” Linear Algebra and Its Applications, vol. 508, pp. 289–300, 2016.
View at: Publisher Site | Google Scholar
C. Y. Deng and H. K. Du, “Representations of the moore-penrose inverse of 2×2 block operator valued matrices,” Journal of the Korean Mathematical Society, vol. 46, no. 6, pp. 1139–1150, 2009.
View at: Publisher Site | Google Scholar
D. E. Ferreyra, F. E. Levis, and N. Thome, “Characterizations of k-commutative equalities for some outer generalized inverses,” Linear and Multilinear Algebra, vol. 68, no. 1, pp. 177–192, 2018.
View at: Publisher Site | Google Scholar
W. Jiang, K. Z. Zuo, and K. Zuo, “Revisiting of the BT-inverse of matrices,” AIMS Mathematics, vol. 6, no. 3, pp. 2607–2622, 2021.
View at: Publisher Site | Google Scholar
K. Zuo and Y. Cheng, “The new revisitation of core EP inverse of matrices,” Filomat, vol. 33, no. 10, pp. 3061–3072, 2019.
View at: Publisher Site | Google Scholar
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 2012.
F. R. Chang and H. C. Chen, “The generalized Cayley-Hamilton theorem for standard pencils,” Systems & Control Letters, vol. 18, no. 3, pp. 179–182, 1992.
View at: Publisher Site | Google Scholar
T. Kaczorek, “An extension of the Cayley-Hamilton theorem for a standard pair of block matrices,” Applied Mathematics and Computation Science, vol. 8, pp. 511–516, 1998.
View at: Google Scholar
T. Kaczorek, “Extension of the Cayley-Hamilton theorem to continuous-time systems with delays,” International Journal of Applied Mathematics and Computer Science, vol. 15, no. 2, pp. 231–234, 2005.
View at: Google Scholar
T. Kaczorek, Selected Problems of Fractional Systems Theory, Springer, Berlin, Germany, 2012.
H. X. Wang, J. L. Chen, and G. J. Yan, “Generalized Cayley-Hamilton theorem for core-EP inverse matrix and DMP inverse matrix,” Journal of Southeast University (English Edition), vol. 34, no. 1, pp. 135–138, 2018.
View at: Google Scholar
T. Kaczorek, “Cayley-Hamilton theorem for Drazin inverse matrix and standard inverse matrices,” Bulletin of the Polish Academy of Sciences, Technical Sciences, vol. 64, no. 4, pp. 793–797, 2016.
View at: Publisher Site | Google Scholar
S. K. Mitra, P. Bhimasankaram, and S. B. Malik, Matrix Partial Orders, Shorted Operators and Applications, World Scientific, Singapore, Singapore, 2010.

Copyright

Copyright © 2021 Na Liu and Hongxing Wang. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

553

Downloads

452

Citations