Gershgorin Circle Theorem

Thorem

• $A ;= (a_{j}^{i}) \in \mathbb{C}^{n \times n}$,
  • matrix
• $D(a_{j}^{i}, R_{i}) \subseteq \mathbb{C}$,
  • circle with radius $R_{i}$ at
• $\lambda_{i} \in \mathbb{C}$,
  • the $i$-th largest eigenvalue of $A$,

\[R_{i} := \sum_{j\neq i} |a_{j}^{i}| .\]

For all $i$,

\[\exists j \in \{1, \ldots, n\}, \text{ s.t. } \lambda_{i} \in D(a_{j}^{j}, R_{j})\]

proof

$\Box$

Proposition

proof

(1)

\[\begin{eqnarray} | \lambda | - | a_{i} | & \le & | \lambda - a_{i} | \nonumber \\ & \le & R_{i} \nonumber \end{eqnarray}\]

Thus

\[\begin{eqnarray} | \lambda | & \le & \sum_{i=1}^{n} |a_{i}| \nonumber \end{eqnarray}\]

$\Box$

Reference

• Gershgorin circle theorem - Wikipedia