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Singular Constrained Linear Systems Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 5, No. 4, 2013 Article ID IJIM-00282, 7 pages Research Article Singular constrained linear systems M. Nikuie ∗ y , M. K. Mirnia z ||||||||||||||||||||||||||||||||{ Abstract In the linear system Ax = b the points x are sometimes constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise in electrical network theory. Keywords : Singular matrix; Drazin inverse; Constrained systems; Bott-Duffin inverse. |||||||||||||||||||||||||||||||||{ 1 Introduction nion equality constrained least squares problem and that of complex equality constrained least squares problem, and obtain a newtechnique inear system of equations play major role in of finding a solution of the quaternion equality L various areas of sciences and engineering constrained least squares problem. such as fuzzy mathematics [1], linear regression [15], etc. There are different forms of linear The constrained linear system system of equations for different purposes such as fuzzy linear system [16, 17], fully fuzzy linear Ax + y = b; x 2 L; y 2 L? system [8, 19], dual linear interval equations [11]. A linear system is consistent if it has a solution, and inconsistent otherwise. In the linear system with given matrix A 2 Cn×n , b 2 Cn and a sub- Ax = b the points x are sometimes constrained space L of Cn is converted to be the equivalent to lie in a given subspace S of column space of linear system of equations [2, 12]. A constrained A. Such system is called a constrained linear linear system that is equivalent to a singular lin- system. By means of complex representation of ear system is called a singular constrained lin- a quaternion matrix, Jianga el al. [9] study the ear system. Constrained linear system arise in relationship between the solutions of the quater- electrical network theory [2, 12]. Electrical net- work has useful in many applications [22]. In ∗ Corresponding author. nikoie [email protected] [2, 7], an electrical network is described topologi- yYoung Researchers and Elite Club, Tabriz Branch, Is- cally in terms of its graph consisting of nodes and lamic Azad University, Tabriz, Iran. zDepartment of Computer engineering, Tabriz Branch, branches, and electrically in terms of its currents Islamic Azad University, Tabriz, Iran. and voltages. 317 318 M. Nikuie /IJIM Vol. 5, No. 4 (2013) 317-323 ( ) x Unlike the case of the nonsingular matrix, and that is a solution of (2.2) if and only if which has a single unique inverse for all purposes, y there are different generalized inverses for differ- ent purposes [2, 5, 21]. Drazin inverse is one x = PLz; y = P ? z = b − APLz; of the generalized inverses and in solving con- L sistent or inconsistent singular linear system has ? been used [2, 14]. There are many methods for where PL orthogonal projector on L, L is called solving linear system using generalized inverses the orthogonal complement of L and z is a solu- [2, 12]. Minimization of quadratic forms using tion of (2.2). the Drazin-inverse solution is investigated in [20]. Definition 2.3 Let A 2 Cn×n and let L be a Bott-Duffin inverse is another generalized inverse n subspace of C . If (APL + PL? ) is nonsingular, that in solving constrained linear system was ap- the Bott-Duffin inverse of A with respect to L, plied [4]. On the consistency of singular con- (−1) denoted by A , is defined by strained systems is discussed in [12]. (L) In this paper, singular constrained linear sys- tem is introduced and solving singular con- (−1) −1 A = P (AP + P ? ) : strained linear system using Drazin inverse is in- (L) L L L vestigated. In Section 2, we recall some prelim- (−1) Some properties of A are collected in [2]. inaries. Section 3 is on the singular constrained (L) system. Solving singular constrained linear sys- Definition 2.4 Let A 2 Cn×n , with ind(A) = k tem is investigated in Section 4. In Section 5 . The matrix X of order n is the Drazin inverse some numerical example, are given, followed by a of A ,denoted by AD , if X satisfies the following suggestion and concluding remarks in Section 6. conditions k k 2 Preliminaries AX = XA; XAX = X; A XA = A : Theorem 2.2 ([5]) ADb is a solution of We first present some preliminaries and basic def- initions which are needed in this paper. For more Ax = b; k = ind(A); (2.3) details, we refer the reader to [2, 5]. if and only if b 2 range(Ak) , and ADb is Definition 2.1 Let A 2 Cn×n . We say the non- an unique solution of (2.3) provided that x 2 negative integer number k to be the index of ma- range(Ak). trix A and is denoted by ind(A), if k is the small- Theorem 2.3 ([5]) Let A 2 Cn×n , with est nonnegative integer number such that ind(A) = k , rank(Ak) = r . We may assume rank(Ak+1) = rank(Ak): that the Jordan normal form of A has the form as follows Definition 2.2 The system ( ) D 0 A = P P −1; Ax + y = b; x 2 L; y 2 L?; (2.1) 0 N where P is a nonsingular matrix, D is a nonsin- with given matrix A 2 Cn×n , b 2 Cn, and a sub- gular matrix of order r , and N is a nilpotent space L of Cn where L? is the orthogonal comple- matrix that N k = 0 . Then we can write the ment of L is called a constrained linear system. Drazin inverse of A in the form ( ) Theorem 2.1 ([2]) The consistency of (2.1) is −1 D 0 − equivalent to the consistency of the following sys- AD = P P 1: 0 0 tem When ind(A) = 1 , it is obvious that N = 0. (APL + PL? )z = b; (2.2) M. Nikuie /IJIM Vol. 5, No. 4 (2013) 317-323 319 3 Singular constrained linear Theorem 3.2 The singular constrained system system (3.4) has a set of solutions if and only if j In this Section, singular constrained linear system rank[APL + PL? ] = rank[APL + PL? b]: is introduced and some results on the singular Proof. We assume that the system (3.4) is constrained linear system, are given. equivalent to the following singular linear system Theorem 3.1 The constrained linear system (APL + PL? )z = b: (3.5) If rank[AP + P ? ] = rank[AP + P ? jb] ? L L L L Ax + y = b; x 2 L; y 2 L ; we know the singular linear system (3.5) has solution. From [10], the singular linear system wherein A is a nonsingular matrix may be equiv- (3.5) has a set of solutions. alent to a singular linear system. Conversely, suppose the system (3.5) is solvable Proof. From [6], we have i.e. j rank(APL) ≤ minfrank(A); rank(PL)g: rank[APL + PL? ] = rank[APL + PL? b]: We know that (APL) is a singular matrix. From From [10], the singular linear system (3.5) has a [6], we have set of solutions. Definition 3.2 The singular constrained linear rank(APL + P ? ) ≤ rank(APL) + rank(P ? ): L L system (3.4) is called a singular consistent con- Therefore it is possible that strained system while rank[APL + P ? ] = rank[APL + P ? jb]; (APL) + rank(PL? ); L L be a singular matrix. inconsistent otherwise. Corollary 3.1 The constrained linear system Corollary 3.2 The equivalent singular linear system of any singular inconsistent constrained Ax + y = b; x 2 L; y 2 L?; system has a set of least-squares solutions. wherein A is a singular matrix may be equivalent Definition 3.3 Let the singular inconsistent to the linear system constrained system Ax + y = b; x 2 L; y 2 L?; (APL + PL? )z = b; is equivalent to the following singular linear sys- wherein (APL + P ? ) is a nonsingular matrix. L tem Definition 3.1 The system (APL + PL? )z = b: The system Ax + y = b; x 2 L; y 2 L?; (3.4) k k (AP + P ? ) (AP + P ? )z = (AP + P ? ) b with given matrix A 2 Cn×n , b 2 Cn, and a L L L L L L n subspace L of C , while (APL + PL? ) be a singu- lar matrix is called a singular constrained linear is called indicial equations wherein k is the index system. of matrix (APL + PL? ). 320 M. Nikuie /IJIM Vol. 5, No. 4 (2013) 317-323 4 Solving singular constrained Proof. According to [18, 13], and properties linear systems of the Drazin inverse [3, 5], in order to obtain the Drazin inverse the projection method solves In this Section, the effect of Drazin inverse in solv- consistent or inconsistent singular linear system ing singular constrained linear system is investi- (4.7) through solving the following indicial gated. equations Theorem 4.1 Let the singular consistent con- k k strained linear system (3.4) be equivalent to the (APL + PL? ) (APL + PL? )z = (APL + PL? ) b: singular linear system (APL + P ? )z = b; (4.6) L Indicial equations is a consistent singular wherein linear system [13], and give a solution using Drazin inverse . Therefore by from [5], z is equal to k = ind(APL + PL? ): k D k ((APL + PL? ) (APL + PL? )) (APL + PL? ) b: A member of set of solutions of the system (4.6) is 5 Numerical examples D (APL + PL? ) b; In this Section, the effect of Drazin inverse in 2 k solving singular constrained linear system is il- if and only if b R((APL + PL? ) ).
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