How to calculate Gershgorin circles of a matrix with complex entries?

Question

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.

Answer 1

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.