Non-Singularity Conditions for Two Z-Matrix Types

ABSTRACT

A real square matrix whose non-diagonal elements are non-positive is called a Z-matrix. This paper shows a necessary and sufficient condition for non-singularity of two types of Z-matrices. The first is for the Z-matrix whose row sums are all non-negative. The non-singularity condition for this matrix is that at least one positive row sum exists in any principal submatrix of the matrix. The second is for the Z-matrix which satisfies where . Let be the ith row and the jth column element of , and be the jth element of . Let be a subset of which is not empty, and be the complement of if is a proper subset. The non-singularity condition for this matrix is such that or such that for . Robert Beauwens and Michael Neumann previously presented conditions similar to these conditions. In this paper, we present a different proof and show that these conditions can be also derived from theirs.

Cite this paper

Miura, S. (2014) Non-Singularity Conditions for Two Z-Matrix Types.*Advances in Linear Algebra & Matrix Theory*, **4**, 109-119. doi: 10.4236/alamt.2014.42009.

Miura, S. (2014) Non-Singularity Conditions for Two Z-Matrix Types.

References

[1] Berman, A. and Plemmons, R.J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, Cambridge.

[2] Ostrowski, A. (1937-38) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96. http://dx.doi.org/10.1007/BF01214284

[3] Varga, R.S. (2000) Matrix Iterative Analysis. 2nd Revised and Expanded Edition, Springer, Berlin.

[4] Nikaido, H. (1968) Convex Structures and Economic Theory. Academic Press, Cambridge.

[5] DeFranza, J. and Gabliardi, D. (2009) Introduction to Linear Algebra with Applications. International Edition, The McGrow-Hill Higher Education.

[6] Anton, H. and Rorres, C. (2011) Elementary Linear Algebra with Supplement Applications. International Student Version,10th Edition, John Wiley & Sons, Boston.

[7] Bretscher, O. (2009) Linear Algebra with Applications. 4th Edition, Pearson Prentice Hall, Upper Saddle River.

[8] Plemmons, R.J. (1976) M-Matrices Leading to Semiconvergent Splittings. Linear Algebra and its Applications, 15, 243-252. http://dx.doi.org/10.1016/0024-3795(76)90030-6

[9] Beauwens, R. (1976) Semistrict Diagonal Dominance. SIAM Journal on Numerical Analysis, 13, 109-112.

http://dx.doi.org/10.1137/0713013

[10] Plemmons, R.J. (1977) M-Matrix Characterizations. 1—Nonsingular M-Matrices. Linear Algebra and Its Applications, 18, 175-188. http://dx.doi.org/10.1016/0024-3795(77)90073-8

[11] Neumann, M. (1979) A Note on Generalizations of Strict Diagonal Dominance for Real Matrices. Linear Algebra and Its Applications, 26, 3-14. http://dx.doi.org/10.1016/0024-3795(79)90168-X

[1] Berman, A. and Plemmons, R.J. (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press, Cambridge.

[2] Ostrowski, A. (1937-38) über die Determinanten mit überwiegender Hauptdiagonale. Commentarii Mathematici Helvetici, 10, 69-96. http://dx.doi.org/10.1007/BF01214284

[3] Varga, R.S. (2000) Matrix Iterative Analysis. 2nd Revised and Expanded Edition, Springer, Berlin.

[4] Nikaido, H. (1968) Convex Structures and Economic Theory. Academic Press, Cambridge.

[5] DeFranza, J. and Gabliardi, D. (2009) Introduction to Linear Algebra with Applications. International Edition, The McGrow-Hill Higher Education.

[6] Anton, H. and Rorres, C. (2011) Elementary Linear Algebra with Supplement Applications. International Student Version,10th Edition, John Wiley & Sons, Boston.

[7] Bretscher, O. (2009) Linear Algebra with Applications. 4th Edition, Pearson Prentice Hall, Upper Saddle River.

[8] Plemmons, R.J. (1976) M-Matrices Leading to Semiconvergent Splittings. Linear Algebra and its Applications, 15, 243-252. http://dx.doi.org/10.1016/0024-3795(76)90030-6

[9] Beauwens, R. (1976) Semistrict Diagonal Dominance. SIAM Journal on Numerical Analysis, 13, 109-112.

http://dx.doi.org/10.1137/0713013

[10] Plemmons, R.J. (1977) M-Matrix Characterizations. 1—Nonsingular M-Matrices. Linear Algebra and Its Applications, 18, 175-188. http://dx.doi.org/10.1016/0024-3795(77)90073-8

[11] Neumann, M. (1979) A Note on Generalizations of Strict Diagonal Dominance for Real Matrices. Linear Algebra and Its Applications, 26, 3-14. http://dx.doi.org/10.1016/0024-3795(79)90168-X