# Sufficient matrix

Let $LaTeX: A$ be an $LaTeX: n \times n$ matrix. Then, $LaTeX: A$ is column sufficient if

$LaTeX: \left [ x_i (Ax)_i \le 0 \mbox{ for all } i \right ] \Rightarrow \left [ x_i (Ax)_i = 0 \mbox{ for all } i \right ].$

$LaTeX: A$ is row sufficient if its transpose is column sufficient. $LaTeX: A$ is sufficient if it is both column and row sufficient. One example is when $LaTeX: A$ is symmetric and positive semi-definite. Here is an example of a matrix that is column sufficient, but not row sufficient:

$LaTeX: A = \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} .$

This arises in linear complementarity problems.