# A NOTE ON PARTIAL SIGN-SOLVABILITY

• Published : 2006.08.01
• 47 8

#### Abstract

In this paper we prove that if AX=b is a partial sign-solvable linear system with A being sign non-singular matrix and if ${\alpha}=\{j:\;x_j\;is\;sign-determined\;by\; Ax=b\}, then$A_{\alpha}X_{\alpha}=b_{\alpha}$is a sign-solvable linear system, where$A_{\alpha}$denotes the submatrix of A occupying rows and columns in o and xo and be are subvectors of x and b whose components lie in${\alpha}$. For a sign non-singular matrix A, let$A_l,\;...,A_{\kappa}$be the fully indecomposable components of A and let${\alpha}_i$denote the set of row numbers of$A_r,\;r=1,\;...,\;k$. We also show that if$A_x=b$is a partial sign-solvable linear system, then, for$r=1,\;...,\;k$, if one of the components of xor is a fixed zero solution of Ax=b, then so are all the components of x_{{\alpha}r}$.

#### Keywords

sign-solvable linear system;partial sign-solvable linear system

#### References

1. L. Bassett, The scope of qualitative economics, Rev. Econ. Studies 29 (1962), 99-132 https://doi.org/10.2307/2295817
2. R. A. Brualdi and B. L. Shader, Matrices of sign-solvable linear systems, Cam bridge University Press, New York, 1995
3. H. Minc, Nonnegative matrices, Wiley, New York, 1988
4. P. A. Samuelson, Foundations of Economic Analysis, Harvard University Press, Cambridge, 1947
5. P. A. Samuelson, Foundations of Economic Analysis, Atheneum, New York, 1971
6. L. Bassett, J. Maybee, and J. Quirk, Qualitative economics and the scope of the correspondence pronciple, Econometrica 36 (1968), 544-563 https://doi.org/10.2307/1909522