一类鞍点矩阵的拟Perron-Frobenius性质
Quasi-Perron-Frobenius property of a class of saddle point matrices
摘要：
许多科学计算领域中出现的鞍点矩阵都具有分块结构$ W= \left(\begin{array}{cc} A & B\\ B^T & C \end{array} \right) $，其中子块$A$是对称正定的，子块$C$是对称半非负定的.在这篇文章中，我们报告了一个不引人注目的，但潜在的理论价值的结论，在某些条件下，特别是当$C$是一个零矩阵，$W$的谱半径必须是最大特征值的$W$。这种性质近似于著名的Perron-Frobenius性质，本文称之为拟Perron-Frobenius性质.在数值试验中，我们观察到了由计算定常斯托克斯方程的混合有限元法导出的鞍点矩阵。数值结果证实了理论分析的正确性，同时也表明使鞍点矩阵具有拟Perron-Frobenius性质的假设条件只是充分条件，而不是必要条件.
英文摘要：
The saddle point matrices arising from many scientific computing fields have block structure $ W= \left(\begin{array}{cc} A & B\\ B^T & C \end{array} \right) $, where the sub-block $A$ is symmetric and positive definite, and $C$ is symmetric and semi-nonnegative definite. In this article we report a unobtrusive but potentially theoretically valuable conclusion that under some conditions, especially when $C$ is a zero matrix, the spectral radius of $W$ must be the maximum eigenvalue of $W$. This characterization approximates to the famous Perron-Frobenius property, and is called quasi-Perron-Frobenius property in this paper. In numerical tests we observe the saddle point matrices derived from some mixed finite element methods for computing the stationary Stokes equation. The numerical results confirm the theoretical analysis, and also indicate that the assumed condition to make the saddle point matrices possess quasi-Perron-Frobenius property is only sufficient rather than necessary.
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