Find the smallest disk in complex plane containing the eigenvalues of a matrix (NBHM 2012) Consider the given matrix
$$\left[\begin{matrix}1& -2& 3& -2 \\1& 1& 0& 3\\-1& 1& 1& -1\\0& -3& 1& 1&\end{matrix}\right]$$
Find out the smallest disk like ($|z-1| < r$ ) in the complex plane containing the eigenvalues of the given matrix.
One easiest method is to find out the eigenvalues from the characteristic equation of the matrix. I can do it so no problem with it. 
I am looking for a method to solve the problem without the above method that can be evaluated in very short time. This type of method is possible, I believe.
Thank you for your help

Using the Gerschgorin Disks approach, we have:


*

*$D_1 = \{|z-1| \le 7\}$

*$D_2 = \{|z-1| \le 4\}$

*$D_3 = \{|z-1| \le 3\}$

*$D_4 = \{|z-1| \le 4\}$

*$D_A = D_1 \cup D_2 \cup D_3\cup  D_4$ - This is the Gerschgorin domain for all of the eigenvalues.


The actual eigenvalues are:


*

*$\lambda_{1,2} = 0.495567~ \pm ~ 3.83314 i$

*$\lambda_{3,4} = 1.50443 ~ \pm ~0.90331 i$

