On nonnegativity of a matrix

水果

This seemingly simple prolbem has troubled me for some time, can anyone share some skill or idea?

luyuanhong

为了便于大家看到这道题，我把它转化为图形文件，显示在下面：

水果

Thank you Lu yuanhong.
When all \theta are equal, the matirx can be regarded as a discretization from a first order partial differential equation.