Question

estimate the eigenvalues of the matrix [1 - 2 3] [6 - 13 18] [4 - 10 14] using the Gershgorin bounds.draw a rough sketch of the region where the eigenvalues lie

Answer 1

Answer on Question #73853 – Math – Quantitative Methods

Question

Estimate the eigenvalues of the matrix

$$\left( \begin{array}{ccc} 1 & -2 & 3 \\ 6 & -13 & 18 \\ 4 & -10 & 14 \end{array} \right)$$

using the Gershgorin bounds. Draw a rough sketch of the region where the eigenvalues lie.

Solution

Gershgorin circle theorem: Every eigenvalue of a square matrix lies in at least one of the Gershgorin discs $C_i$. The possible range of the eigenvalues is defined by the outer borders of the union of all discs

$$C = \bigcup_{i=1}^{n} C_i$$

where

$$C_i = \{|c - a_{ii}| \leq r_i \}$$ $$r_i = \sum_{\substack{j=1 \\ j \neq i}}^{n} |a_{ij}|$$

There are three Gershgorin discs in this matrix:

- $C_1$ with the centre point $a_{11} = 1$ and radius $r_1 = 2 + 3 = 5$

- $C_2$ with the centre point $a_{22} = -13$ and radius $r_2 = 6 + 18 = 24$

- $C_3$ with the centre point $a_{33} = 14$ and radius $r_3 = 4 + 14 = 18$

Answer: region where the eigenvalues lie:

$$C _ {1} \cup C _ {2} \cup C _ {3}$$

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