Abstract

We will consider Rohde's general form of -inverse of a matrix . The necessary and sufficient condition for consistency of a linear system will be represented. We will also be concerned with the minimal number of free parameters in Penrose's formula for obtaining the general solution of the linear system. These results will be applied for finding the general solution of various homogenous and nonhomogenous linear systems as well as for different types of matrix equations.

1. Introduction

In this paper, we consider nonhomogeneous linear system in variables where is an matrix over the field of rank and is an matrix over . The set of all matrices over the complex field will be denoted by , . The set of all matrices over the complex field of rank will be denoted by . For simplicity of notation, we will write () for the th row (the th column) of the matrix .

Any matrix satisfying the equality is called -inverse of and is denoted by . The set of all -inverses of the matrix is denoted by . It can be shown that is not empty. If the matrix is invertible, then the equation has exactly one solution , so the only -inverse of the matrix is its inverse ; that is, =. Otherwise, -inverse of the matrix is not uniquely determined. For more information about -inverses and various generalized inverses, we recommend Ben-Israel and Greville [1] and Campbell and Meyer [2].

For each matrix there are regular matrices and such that where is identity matrix. It can be easily seen that every -inverse of the matrix can be represented in the form where , , and are arbitrary matrices of corresponding dimensions , , and with mutually independent entries; see Rohde [3] and Perić [4].

We will generalize the results of Urquhart [5]. Firstly, we explore the minimal numbers of free parameters in Penrose's formula for obtaining the general solution of the system (1). Then, we consider relations among the elements of to obtain the general solution in the form of the system (1) for . This construction has previously been used by Malešević and Radičić [6] (see also [7] and [8]). At the end of this paper, we will give an application of this results to the matrix equation .

2. The Main Result

In this section, we indicate how a technique of an -inverse may be used to obtain the necessary and sufficient condition for an existence of a general solution of a nonhomogeneous linear system.

Lemma 1. The nonhomogeneous linear system (1) has a solution if and only if the last coordinates of the vector are zeros, where is regular matrix such that (2) holds.

Proof. The proof follows immediately from Kronecker-Capelli theorem. We provide a new proof of the lemma by using the -inverse of the system matrix . The system (1) has a solution if and only if ; see Penrose [9]. Since is described by (3), it follows that Hence, we have the following equivalences: Furthermore, we conclude that .

Theorem 2. The vector   is an arbitrary column, is the general solution of the system (1), if and only if the -inverse of the system matrix has the form (3) for arbitrary matrices and and the rows of the matrix are free parameters, where and .

Proof. Since -inverse of the matrix has the form (3), the solution of the system can be represented in the form According to Lemma 1 and from (2), we have Furthermore, we obtain where . We now conclude that Therefore, since matrix is regular, we deduce that is the general solution of the system (1) if and only if the rows of the matrix are free parameters.

Corollary 3. The vector is an arbitrary column, is the general solution of the homogeneous linear system , , if and only if the -inverse of the system matrix has the form (3) for arbitrary matrices and and the rows of the matrix are free parameters, where .

Example 4. Consider the homogeneous linear system The system matrix is For regular matrices equality (2) holds. Rohde's general -inverse of the system matrix is of the form According to Corollary 3 the general solution of the system (13) is of the form where Therefore, we obtain If we take as a parameter, we get the general solution

Corollary 5. The vector is the general solution of the system (1), if and only if the -inverse of the system matrix has the form (3) for arbitrary matrices and and the rows of the matrix are free parameters, where .

Remark 6. Similar result can be found in the paper by Malešević and Radičić [6].

Example 7. Consider the nonhomogeneous linear system According to Corollary 5, the general solution of the system (22) is of the form If we take as a parameter, we obtain the general solution of the system
We are now concerned with the matrix equation where , , and .

Lemma 8. The matrix equation (25) has a solution if and only if the last rows of the matrix are zeros, where is regular matrix such that (2) holds.

Proof. If we write and , then we can observe the matrix equation (25) as the system of matrix equations Each of the matrix equation , , by Lemma 1 has solution if and only if the last coordinates of the vector are zeros. Thus, the previous system has solution if and only if the last rows of the matrix are zeros, which establishes that matrix equation (25) has solution if and only if all entries of the last rows of the matrix are zeros.

Theorem 9. The matrix is an arbitrary matrix, is the general solution of the matrix equation (25) if and only if the -inverse of the system matrix has the form (3) for arbitrary matrices and and the entries of the matrix are mutually independent free parameters, where and .

Proof. Applying Theorem 2 on each system , , we obtain that is the general solution of the system if and only if the rows of the matrix are free parameters. Assembling these individual solutions together we get that is the general solution of matrix equation (25) if and only if entries of the matrix are mutually independent free parameters.

From now on we proceed with the study of the nonhomogeneous linear system of the form where is an matrix over the field of rank and is a matrix over . Let and let be regular matrices such that An -inverse of the matrix can be represented in Rohde's form where , , and are arbitrary matrices of corresponding dimensions , and with mutually independent entries.

Lemma 10. The nonhomogeneous linear system (31) has a solution if and only if the last elements of the row are zeros, where is regular matrix such that (32) holds.

Proof. By transposing the system (31), we obtain system and by transposing matrix equation (32) we obtain that . According to Lemma 1, the system has solution if and only if the last coordinates of the vector are zeros, that is, if and only if the last elements of the row are zeros.

Theorem 11. The row is an arbitrary row, is the general solution of the system (31), if and only if the -inverse of the system matrix has the form (33) for arbitrary matrices and and the columns of the matrix are free parameters, where and .

Proof. The basic idea of the proof is to transpose the system (31) and to apply Theorem 2. The -inverse of the matrix is equal to a transpose of the -inverse of the matrix . Hence, we have We can now proceed analogously to the proof of Theorem 2 to obtain that is the general solution of the system if and only if the rows of the matrix are free parameters. Therefore, is the general solution of the system (31) if and only if the columns of the matrix are free parameters. Analogous corollaries hold for Theorem 11.

We now deal with the matrix equation where , , and .

Lemma 12. Matrix equation (38) has a solution if and only if the last columns of the matrix are zeros, where is regular matrix such that (32) holds.

Theorem 13. The matrix is an arbitrary matrix, is the general solution of the matrix equation (38) if and only if the -inverse of the system matrix has the form (33) for arbitrary matrices and and the entries of the matrix are mutually independent free parameters, where and .

3. An Application

In this section we will briefly sketch properties of the general solution of matrix equation where , , , and . If we denote by matrix product , then the matrix equation (41) becomes According to Theorem 9, the general solution of the system (42) can be presented as a product of the matrix and the matrix which has the first rows same as the matrix and the elements of the last rows are mutually independent free parameters; and are regular matrices such that . Thus, we are now turning on to the system of the form By Theorem 13, we conclude that the general solution of the system (43) can be presented as a product of the matrix which has the first columns equal to the first columns of the matrix and the rest of the columns have mutually independent free parameters as entries, and the matrix , for regular matrices and such that . Therefore, the general solution of the system (41) is of the form where is a submatrix of the matrix corresponding to the first rows and the first columns and the entries of the matrices , , and are free parameters. We will illustrate this on the following example.

Example 14. We consider the matrix equation where , and . If we take , we obtain the system It is easy to check that the matrix is of the rank and for matrices and the equality holds. Based on Theorem 9, the equation can be rewritten in the system form Combining Theorem 2 with the equality yields for an arbitrary matrix . Therefore, the general solution of the system is From now on, we consider the system for There are regular matrices and such that holds. Since the rank of the matrix is , according to Lemma 12 all entries of the last two columns of the matrix are zeros; that is, we have , . Hence, we get that the matrix is of the form . Applying Theorem 13, we obtain for an arbitrary matrix . Finally, the solution of the system is