Given a diagonal matrix $D$ (with sum of diagonal elements be negative), a row-stochastic $W_r$ (right eigenvector $\mathbf{1}$ with eigenvalue 1), and column-stochastic $W_c$ (left eigenvector $\mathbf{1}$ with eigenvalue 1), how to prove that below matarix has spectral radius greater than one? ($W_r$ and $W_c$ are irreducible) $$ \begin{bmatrix} W_r-D & - I \\ (W_c-I)D &W_c \end{bmatrix}. $$ I am pretty sure it is true but cannot prove it. I also ran also many simulations which confirmed my conjecture.
I really appreciate any help. Thanks.
I am not sure if this is enough for you, but it is easy to see that the spectral radius is greater than or equal to one, and all we need is that $W_r$ is row-stochastic. The matrix has eigenvalue one $$ \begin{bmatrix}W_r-D & -I\\(W_c-I)D & W_c\end{bmatrix} \begin{bmatrix}{\mathbf 1}\\-D{\mathbf 1}\end{bmatrix}= \begin{bmatrix}{\mathbf 1}\\-D{\mathbf 1}\end{bmatrix}. $$