Sufficient matrix

From Glossary

Jump to: navigation, search

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

\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:

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

This arises in linear complementarity problems.

Personal tools