  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.
