A NOTE ON PARTIAL SIGN-SOLVABILITY

Title & Authors
A NOTE ON PARTIAL SIGN-SOLVABILITY
Hwang, Suk-Geun; Park, Jin-Woo;

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 $\small{{\alpha}=\{j:\;x_j\;is\;sign-determined\;by\; Ax=b\}}$, then $\small{A_{\alpha}X_{\alpha}=b_{\alpha}}$ is a sign-solvable linear system, where $\small{A_{\alpha}}$ 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 $\small{{\alpha}}$. For a sign non-singular matrix A, let $\small{A_l,\;...,A_{\kappa}}$ be the fully indecomposable components of A and let $\small{{\alpha}_i}$ denote the set of row numbers of $\small{A_r,\;r=1,\;...,\;k}$. We also show that if $\small{A_x=b}$ is a partial sign-solvable linear system, then, for $\small{r=1,\;...,\;k}$, if one of the components of xor is a fixed zero solution of Ax=b, then so are all the components of $\small{x_{{\alpha}r}}$.
Keywords
sign-solvable linear system;partial sign-solvable linear system;
Language
English
Cited by
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