I am trying to approximate or calculate the eigenvalues of this
matrix.$$\begin{bmatrix}
4 & 1+i & 0 & 0 \\
1-i & 3 & -1 & 0 \\
0 & -1 & -2 & 0.1 \\
0 & 0 & 0.1 & 5 \\
\end{bmatrix}$$How to approximate the eigenvalues using Gershgorin circle Theorem if
the sum value of the non diagonal value in a row has an imaginary part
like this?
As far as I know, the radius of the Gershgorin circle can be determined as the sum value of the non diagonal value in a row.
Best Answer
You have to sum the absolute values. Consider for example the Gershgorin circle $\bar S_1$, which is associated to $a_{11} = 4$ and has, by definition, the form
$$\bar S_1 = \bar S \bigg(4, \sum_{j = 2}^4 \lvert a_{1j} \rvert \bigg).$$
We have
$$\sum_{j = 2}^4 \lvert a_{1j} \rvert = \lvert 1 + \mathrm i \rvert + 0 + 0 = \sqrt{2}, $$
hence
$$\bar S_1 = \bar S(4, \sqrt{2}).$$
The other circles can be calculated similar.