- #1

mathmari

Gold Member

MHB

- 5,049

- 7

We have the matrix $$A=\begin{pmatrix}2 & 0.4 & -0.1 & 0.3 \\ 0.3 & 3 & -0.1 & 0.2 \\ 0 & 0.7 & 3 & 1 \\ 0.2 & 0.1 & 0 & 4\end{pmatrix}$$ We get the row Gershgorin circles: $$K_1=\{z\in \mathbb{C} : |z-2|\leq 0.8 \} \\ K_2=\{z\in \mathbb{C} : |z-3|\leq 0.6 \} \\ K_3=\{z\in \mathbb{C} : |z-3|\leq 1.7 \} \\ K_4=\{z\in \mathbb{C} : |z-4|\leq 0.3 \} $$ and the column Gershgorin circles: $$K_1'=\{z\in \mathbb{C} : |z-2|\leq 0.5 \} \\ K_2'=\{z\in \mathbb{C} : |z-3|\leq 1.2 \} \\ K_3'=\{z\in \mathbb{C} : |z-3|\leq 0.2 \} \\ K_4'=\{z\in \mathbb{C} : |z-4|\leq 1.5 \} $$

To get the intervals of the eigenvalues, we have to look at the union of all row circles and the union of all column circles, or not?

Then at the result we have to take the intersection, or not?

Till now I have seen only examples where $A=A^T$ and so the row and column circles are the same, and so I am confused here.

(Wondering)