I want to prove the extension of the Gerschgorin Circle Theorem (called the Ostrowski disk theorem) below:
Prove that any eigenvalue of matrix $A$ must locates in some of the disks: $$G_{i;\tau}(A)=\{\zeta:|\zeta-\alpha_{ii}|\le(\sum_{j\neq i}|\alpha_{ij}|)^\tau(\sum_{j\neq i}|\alpha_{ji}|)^{1-\tau})\}$$ for any $\tau\in[0,1]$.
This is slightly stronger than the original Gerschgorin Circle Theorem. And I worked out that the above theorem is actually equivalent to changing the $G_{i;\tau}(A)$ into $$G_{i,j;\tau}(A):=\{\zeta:|\zeta-\alpha_{ii}||\zeta-\alpha_{jj}|\le(\sum_{k\neq i}|\alpha_{ik}|)^\tau(\sum_{k\neq i}|\alpha_{ki}|)^{1-\tau}(\sum_{k\neq j}|\alpha_{jk}|)^\tau(\sum_{k\neq j}|\alpha_{kj}|)^{1-\tau}\}$$ however, I don't know which theorem above is easier and I don't have further ideas of proving (We can use Bauer-Fike theorem to prove the Gerschgorin Circle Theorem, but I don't know how to use Bauer-Fike to prove this), are there any hints of proving any of the theorems above? Thanks!