linear algebra - Maximum eigenvalue of a special $m$-matrix

I have a set of matrix, which is:

1. Real symmetric positive definite. Very sparse.


2. Diagonal elements are positive while off-diagonal elements are negative.


3. $\displaystyle a_{ii}=-\sum^{n}_{{j=1}\atop{j\ne i}} a_{ij}$


4. $a_{ii} \in (0,1]$


5. $a_{ij} \in (-1,0]$ when $i \neq j$

My experiments show that the largest eigenvalue of all the matrices I have are larger than 1. Can some one help me on proving that $\lambda_{max} >1$ for this matrix?

My first though is to prove that $Ax=\lambda x < x$ does not hold. But I couldn't get any break through. Thanks!