P-matrix

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A square matrix with all of its principal minors positive. This includes all symmetric, positive definite matrices. Here is an example of a P-matrix that is not positive definite:


LaTeX: 
A = \begin{bmatrix}
1 & -3 \\
0 & 1
\end{bmatrix}


The principal minors are positive, but LaTeX: \textstyle (1, 1)A(1, 1)^t < 0. The significance of this class is in the theorem:

The linear complementarity problem, defined by LaTeX: (M, q), has a unique solution for each q in Rn if, and only if, LaTeX: M is a P-matrix.


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