In this section, we discuss the non-singularity of the NSZ-matrix. A Necessary and Sufficient Condition for Non-Singularity of the Z-Matrix Whose Row Sums Are All Non-Negative is itself.īased on the above, we confirm the following basic proposition.Ģ. Similarly, refers to a submatrix of whose row elements belong to and column elements belong to, and refers to a submatrix of whose row and column elements belong to. refers to a submatrix of whose row elements belong to and column elements belong to. When is a proper subset, we define as the complement of. refers to a submatrix of whose row and column elements belong to. Let be a set of number of rows and columns of a square matrix and let be a subset of which is not empty. Here, we define the notation for submatrices in this paper. Theorem 1.7 Similar matrices have the same eigenvalue 6. Theorem 1.6 When two matrices are similar, if one matrix is non-singular, the other is also non-singular. If holds for the square matrices and and a non-singular matrix, and are called similar to each other. The determinant of a matrix whose column vectors are linearly dependent is zero 5. Theorem 1.5 If the row sums of a square matrix are all zeroes. Regarding the inverse of an M-matrix, is satisfied 4. However, as it receives less attention that diagonal elements of are or more, we confirm this aspect. As Theorems 1.3 and 1.4 are well known, we entrust the proof to another work 3. In particular, all diagonal elements of the inverse are equal to or more than 2. In this case, and all elements of the inverse of an M-matrix are non-negative. Theorem 1.4 An M-matrix is non-singular if and only if. If a Z-matrix satisfies, is called an M-matrix 1. Then there exists a non-negative eigenvector corresponding to. Let be the maximum non-negative eigenvalue of. Theorem 1.3 A non-negative square matrix always has a non-negative eigenvalue. On the other hand, the diagonal elements of B are. Take a real number which is equal to or more than all diagonal elements and construct the matrix where refers to the unit matrix. The determinant of a square matrix is denoted in this paper. We first state the basic propositions of linear algebra used in this paper. However, we will prove the latter condition first, and then address the former condition. Therefore, if we can find a necessary and sufficient condition for non-singularity of the NPZ-matrix, we find the necessary and sufficient condition for non-singularity of the NSZ-matrix automatically. Thus,Īs Theorem 1.1 shows, the NSZ-matrix is a type of the NPZ-matrix. If we divide both sides of by, we obtain for. Hence, for is satisfied.Ĭonversely, consider that is an NPZ-matrix which satisfies for where. If is an NSZ-matrix, the ith element of is. Let be a set of numbers, and be a positive vector with all elements equal to the same number. Theorem 1.1 An NSZ-matrix is equivalent to an NPZ-matrix where all elements of are the same number. The following relation exists between these matrices. In this paper, we denote this as a Non-negative Product Z-matrix (NPZ-matrix). The second is the Z-matrix A which satisfies where. In this paper, we denote this as a Non-negative Sums Z-matrix (NSZ-matrix). The first is the Z-matrix whose row sums are all non-negative. The purpose of this paper is to show a necessary and sufficient condition for non-singularity of two types of Z-matrices. Keywords:Z-Matrix, M-Matrix, Non-Negative Matrix, Diagonal DominanceĪ real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. In this paper, we present a different proof and show that these conditions can be also derived from theirs. Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. The non-singularity condition for this matrix is such that or such that for. Let be a subset of which is not empty, and be the complement of if is a proper subset. Let be the ith row and the jth column element of, and be the jth element of. The second is for the Z-matrix which satisfies where. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The first is for the Z-matrix whose row sums are all non-negative. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. Received 23 April 2014 revised accepted Ī real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This work is licensed under the Creative Commons Attribution International License (CC BY). Email: © 2014 by author and Scientific Research Publishing Inc.
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