Research Article  Open Access
Xiaoji Liu, Guangyan Zhu, Guangping Zhou, Yaoming Yu, "An Analog of the Adjugate Matrix for the Outer Inverse ", Mathematical Problems in Engineering, vol. 2012, Article ID 591256, 14 pages, 2012. https://doi.org/10.1155/2012/591256
An Analog of the Adjugate Matrix for the Outer Inverse
Abstract
We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse . We show the equivalent relationship between the existence and limiting expression of and some limiting processes of matrices and deduce the new determinantal representations of , based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation .
1. Introduction
Throughout this paper denotes the set of matrices over the complex number field , and denotes its subset in which every matrix has rank . stands for the identity matrix of appropriate order (dimension).
Let , and let and be Hermitian positive definite matrices of orders and , respectively. Consider the following equations: is called a  (or outer) inverse of if it satisfies (2) and denoted by . is called the MoorePenrose inverse of if it satisfies (1), (2), (3), and (5) and denoted by . is called the weighted MoorePenrose inverse of (with respect to ) if it satisfies (1), (2), (4), and (6) and denoted by (see, e.g., [1, 2]).
Let . Then a matrix satisfying where is some positive integer, is called the Drazin inverse of and denoted by . The smallest positive integer such that and satisfy (7), (8), and (9), then it is called the Drazin index and denoted by . It is clear that is the smallest positive integer satisfying (see [3]). If , then is called the group inverse of and denoted by . As is well known, exists if and only if . The generalized inverses, and in particular MoorePenrose, group and Drazin inverses, have also been studied in the context of semigroups, rings of Banach and algebras (see [4–8]).
In addition, if a matrix satisfies (1) and (5), then it is called a inverse of and is denoted by .
Let , . Then the matrix satisfying where is some nonnegative integer, is called the weighted Drazin inverse of , and is denoted by (see [9]). It is obvious that when and , is called the Drazin inverse of .
Lemma 1.1 (see [1, Theorem 2.14]). Let , and let and be subspaces of and , respectively, with . Then has a inverse such that and if and only if in which case is unique and denoted by .
If exists and there exists a matrix such that and , then and .
It is well known that several important generalized inverses, such as the MoorePenrose inverse , the weighted MoorePenrose inverse , the Drazin inverse , and the group inverse , are outer inverses for some specific choice of and , are all the generalized inverse ,  (or outer) inverse of with the prescribed range and null space (see [2, 10] in the context of complex matrices and [11] in the context of semigroups).
Determinantal representation of the generalized inverse was studied in [12, 13]. We will investigate further such representation by exploiting the limiting expression for . The paper is organized as follows. In Section 2, we investigate the equivalent relationship between the existence of and the limiting process of matrices or and deduce the new determinantal representations of , based on some analog of the classical adjoint matrix, by exploiting limiting expression. In Section 3, using the analog of the classical adjoint matrix in Section 2, we present Cramer rules for the restricted matrix equation . In Section 4, we give an example for solving the solution of the restricted matrix equation by using our expression. We introduce the following notations.
For , the symbol denotes the set . And , where .
Let . The symbols and stand for the th column and the th row of , respectively. In the same way, denote by and the th column and the th row of Hermitian adjoint matrix . The symbol (or ) denotes the matrix obtained from by replacing its th column (or row) with some vector (or ). We write the range of by and the null space of by . Let . We define the range of a pair of and as .
Let and , where . Then denotes a minor of determined by the row indexed by and the columns indexed by . When , the cofactor of in is denoted by .
2. Analogs of the Adjugate Matrix for
We start with the following theorem which reveals the intrinsic relation between the existence of and of or . Here means through any neighborhood of 0 in which excludes the nonzero eigenvalues of a square matrix. In [14], Wei pointed out that the existence of implies the existence of or . The following result will show that the converse is true under some condition.
Theorem 2.1. Let , and let and be subspaces of and , respectively, with . Let with and . Then the following statements are equivalent:(i) exists;(ii) exists and ;(iii) exists and .
In this case,
Proof. (i)⇔(ii) Assume that exists. By [14, Theorem 2.4], exists. Since , .
Conversely, assume that exists and . So
exists. By [15, Theorem], exists. So exists, and then, by [13, Theorem 2], exists.
Similarly, we can show that (i)⇔(iii). Equation (2.1) comes from [14, equation (2.16)].
Lemma 2.2. Let and . Then , where , , and , where , .
Proof. Let be an matrix with in the entry, 1 in all diagonal entries, and 0 in others. It is an elementary matrix and
It follows from the invertibility of , that .
Analogously, the inequation can be proved. So the proof is complete.
Recall that if is the characteristic polynomial of an matrix— over , then is the sum of all principal minors of , where (see, e.g., [16]).
Theorem 2.3. Let , and be the same as in Theorem 2.1. Write . Suppose that the generalized inverse of exists. Then can be represented as follows: where or where
Proof. We will only show the representation (2.5) since the proof of (2.7) is similar. If is not the eigenvalue of , then the matrix is invertible, and
where , are cofactors of . It is easy to see that
So, by (2.1),
We have the characteristic polynomial of
where is a sum of principal minors of . Since , and
Expanding , we have
where , , for and .
By Lemma 2.2, and so , and , for all . Therefore, , , for all . Consequently,
Substituting (2.12) and (2.14) into (2.10) yields
Substituting for in the above equation, we reach (2.5).
Remark 2.4. The proofs of Lemma 2.2 and Theorem 2.3 are based on the general techniques and methods obtained previously by [17], respectively.
Remark 2.5. (i) By using (2.5), we can obtain (2.17) in [12, Theorem 2.3]. In fact, and, by the BinetCauchy formula,
where . Note that if or . In addition, using the symbols in [13], we can rewrite (2.5) as [13, equation (13)] over .
(ii) This method is especially efficient when or is given (comparing with [12, Theorem 2]).
Observing the particular case from Theorem 2.3, , where and are Hermitian positive definite matrices, we obtain the following corollary in which the symbols and .
Corollary 2.6. Let and , where and are Hermitian positive definite matrices of order and , respectively, Then where or where
If and are identity matrices, then we can obtain the following result.
Corollary 2.7 (see [17, Theorem 2.2]). The MoorePenrose inverse of can be represented as follows: where or where
Note that . Therefore, when in Theorem 2.3, we have the following corollary.
Corollary 2.8. Let , , and . If , , and , then where or where
When with in Theorem 2.3, we have the following corollary.
Corollary 2.9 (see [17, Theorem 3.3]). Let with and , and . Then where
Finally, we turn our attention to the two projectors and . The limiting expressions for in (2.1) bring us the following:
Corollary 2.10. Let , and be the same as in Theorem 2.1. Write and . Suppose that exists. Then of can be represented as follows: where where
3. Cramer Rules for the Solution of the Restricted Matrix Equation
The restricted matrix equation problem is mainly to find solution of a matrix equation or a system of matrix equations in a set of matrices which satisfy some constraint conditions. Such problems play an important role in applications in structural design, system identification, principal component analysis, exploration, remote sensing, biology, electricity, molecular spectroscopy, automatics control theory, vibration theory, finite elements, circuit theory, linear optimal, and so on. For example, the finiteelement static model correction problem can be transformed to solve some constraint condition solution and its best approximation of the matrix equation . The undamped finiteelement dynamic model correction problem can be attributed to solve some constraint condition solution and its best approximation of the matrix equation . These motivate the gradual development of theory in respect of the solution to the restricted matrix equation in recent years (see [18–27]).
In this section, we consider the restricted matrix equation where , , , , , and , satisfy Assume that there exist matrices and satisfying If and exist and , then the restricted matrix equation (3.1) has the unique solution (see [2, Theorem 3.3.3] for the proof).
In particular, when is a vector and , the restricted matrix equation (3.1) becomes the restricted linear equation If , then is the unique solution of the restricted linear equation (3.5) (see also [10, Theorem 2.1]).
Theorem 3.1. Given , and as above. If and exist and , then is the unique solution of the restricted matrix equation (3.1) and it can be represented as or where and , , and .
Proof. By the argument above, we have is the unique solution of the restricted matrix equation (3.1). Setting and using (2.7), we get that
where . Since , by (2.5),
Hence, we have (3.6).
We can obtain (3.7) in the same way.
In particular, when is a vector and in the above theorem, we have the following result from (3.7).
Theorem 3.2. Given , and as above. If , then is the unique solution of the restricted linear equation , , and it can be represented as where .
Remark 3.3. Using the symbols in [13], we can rewrite (3.10) as [13, equation (27)].
4. Example
Let Obviously, , , and It is easy to verify that and . Thus, and exist by Lemma 1.1.
Now consider the restricted matrix equation Clearly, and it is easy to verify that and hold.
Note that and if and only if . So, by Theorem 3.1, the unique solution of (4.3) exists.
Computing and setting , we have Table 1.

Similarly, setting , we have Table 2.

So, by (3.7), we have
Acknowledgments
The authors would like to thank the referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China 11061005, the Ministry of Education Science and Technology Key Project under Grant 210164, and Grants HCIC201103 of Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis Open Fund.
References
 A. BenIsrael and T. N. E. Greville, Generalized Inverse Theory and Application, SpringerVerlag, New York, NY, USA, 2nd edition, 2003.
 G. Wang, Y. Wei, and S. Qiao, Generalized Inverse: Theory and Computations, Science Press, Beijing, China, 2004.
 M. P. Drazin, “Pseudoinverses in associative rings and semigroups,” The American Mathematical Monthly, vol. 65, pp. 506–514, 1958. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 P. Patricio and R. Puystjens, “Drazinmoorepenrose invertibility in rings,” Linear Algebra and its Applications, vol. 389, pp. 159–173, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 D. Huang, “Group inverses and Drazin inverses over Banach algebras,” Integral Equations and Operator Theory, vol. 17, no. 1, pp. 54–67, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 R. Harte and M. Mbekhta, “On generalized inverses in ${C}^{\ast}$algebras,” Studia Mathematica, vol. 103, no. 1, pp. 71–77, 1992. View at: Google Scholar
 P. S. Stanimirović and D. S. CvetkovićIlić, “Successive matrix squaring algorithm for computing outer inverses,” Applied Mathematics and Computation, vol. 203, no. 1, pp. 19–29, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Miladinović, S. Miljković, and P. Stanimirović, “Modified SMS method for computing outer inverses of Toeplitz matrices,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3131–3143, 2011. View at: Publisher Site  Google Scholar
 R. E. Cline and T. N. E. Greville, “A Drazin inverse for rectangular matrices,” Linear Algebra and its Applications, vol. 29, pp. 54–62, 1980. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. L. Chen, “A Cramer rule for solution of the general restricted linear equation,” Linear and Multilinear Algebra, vol. 34, no. 2, pp. 177–186, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 X. Mary, “On generalized inverses and Green's relations,” Linear Algebra and its Applications, vol. 434, no. 8, pp. 1836–1844, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. Yu and G. Wang, “On the generalized inverse ${A}_{T,S}^{(2)}$ over integer domains,” The Australian Journal of Mathematical Analysis and Applications, vol. 4, no. 1, article 16, pp. 1–20, 2007. View at: Google Scholar
 Y. Yu and Y. Wei, “Determinantal representation of the generalized inverse ${A}_{T,S}^{(2)}$ over integral domains and its applications,” Linear and Multilinear Algebra, vol. 57, no. 6, pp. 547–559, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Y. Wei, “A characterization and representation of the generalized inverse ${A}_{T,S}^{(2)}$ and its applications,” Linear Algebra and its Applications, vol. 280, no. 23, pp. 87–96, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 A. BenIsrael, “On matrices of index zero or one,” Society for Industrial and Applied Mathematics Journal on Applied Mathematics, vol. 17, pp. 1118–1121, 1969. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 R. A. Horn and C. R. Johnson, The Matrix analysis, Cambridge University Press, New York, NY, USA, 1985.
 I. I. Kyrchei, “Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules,” Linear and Multilinear Algebra, vol. 56, no. 4, pp. 453–469, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. Baruch, “Optimal correction of mass and stiffness matrices usingmeasured modes,” American Institute of Aeronautics and Astronautics Journal, vol. 20, no. 11, pp. 1623–1626, 1982. View at: Publisher Site  Google Scholar
 A. Bjierhammer, “Rectangular reciprocal matrices with special reference to geodetic calculations,” Kungliga tekniska hogskolan Handl Stackholm, vol. 45, pp. 1–86, 1951. View at: Google Scholar
 M. Bixon and J. Jortner, “Intramolecular radiationless transitions,” Journal of Chemical and Physics, vol. 48, pp. 715–726, 1968. View at: Google Scholar
 J. W. Gadzuk, “Localized vibrational modes in Fermi liquids. General theory,” Physical Review B, vol. 24, no. 4, pp. 1651–1663, 1981. View at: Publisher Site  Google Scholar
 L. Datta and S. D. Morgera, “On the reducibility of centrosymmetric matrices applications in engineering problems,” Circuits, Systems, and Signal Processing, vol. 8, no. 1, pp. 71–96, 1989. View at: Publisher Site  Google Scholar
 Y. M. Ram and G. G. L. Gladwell, “Contructing a finite elemnt model of a vibratory rod from eigendata,” Journal of Sound and Vibration, vol. 169, no. 2, pp. 229–237, 1994. View at: Publisher Site  Google Scholar
 A. Jameson and E. Kreindler, “Inverse problem of linear optimal control,” Society for Industrial and Applied Mathematics Journal on Control and Optimization, vol. 11, pp. 1–19, 1973. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 G. W. Stagg and A. H. ElAbiad, Computer Methodes in Power System Analysis, MeGrawHill, New York, NY, USA, 1968.
 H. C. Chen, “The SAS domain decomposition method for structural anlaysis,” CSRD Technical Report 754, Center for Supercomputing Research and Development. University of Illinois, Urbana, Ill, USA, 1988. View at: Google Scholar
 F. S. Wei, “Analytical dynamic model improvement using vibration test data,” American Institute of Aeronautics and Astronautics Journal, vol. 28, no. 1, pp. 175–177, 1990. View at: Google Scholar
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