Sufficient matrix

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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.


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