Proof of the Gershgorin Circle Theorem (Detailed)

The Gershgorin theorem provides a simple yet powerful method for eigenvalue localization. We present the full proof including the component-counting refinement and column variation.

Statement

Theorem6.3Gershgorin Theorem with Refinements

Let $A = (a_{ij}) \in \mathbb{C}^{n \times n}$ . Define row discs $D_i^R = \{z : |z - a_{ii}| \leq R_i\}$ with $R_i = \sum_{j\neq i}|a_{ij}|$ , and column discs $D_j^C = \{z : |z - a_{jj}| \leq C_j\}$ with $C_j = \sum_{i\neq j}|a_{ij}|$ . Then:

Every eigenvalue lies in $\bigcup_i D_i^R$ and also in $\bigcup_j D_j^C$ .
Every eigenvalue lies in $\bigcup_i (D_i^R \cap D_i^C)$ is not true in general, but every eigenvalue lies in the intersection $\left(\bigcup_i D_i^R\right) \cap \left(\bigcup_j D_j^C\right)$ .
If a union of $m$ discs is disjoint from the remaining $n - m$ discs, it contains exactly $m$ eigenvalues.

Proof

Row discs. Let $(\lambda, x)$ be an eigenpair with $\|x\|_\infty = |x_p| = 1$ . From $Ax = \lambda x$ : $\sum_j a_{pj}x_j = \lambda x_p$ , giving $(\lambda - a_{pp})x_p = \sum_{j \neq p} a_{pj}x_j$ . Thus $|\lambda - a_{pp}| \leq \sum_{j \neq p}|a_{pj}||x_j|/|x_p| \leq R_p$ .

Column discs. Since $\lambda$ is also an eigenvalue of $A^T$ (same characteristic polynomial), applying the row-disc result to $A^T$ gives $\lambda \in D_j^C$ for some $j$ , proving statement (2).

Component counting (detailed). Define $A(t) = (1-t)D + tA$ for $t \in [0,1]$ where $D = \mathrm{diag}(A)$ . The off-diagonal entries of $A(t)$ are $t \cdot a_{ij}$ for $i \neq j$ , so the Gershgorin radii for $A(t)$ are $R_i(t) = t R_i$ .

The eigenvalues $\lambda_1(t), \ldots, \lambda_n(t)$ of $A(t)$ are continuous functions of $t$ (they are roots of $\det(A(t) - \lambda I) = 0$ , a polynomial in $\lambda$ with continuous coefficients).

At $t = 0$ : $\lambda_i(0) = a_{ii}$ and $D_i(0) = \{a_{ii}\}$ . Exactly one eigenvalue starts in each disc (assuming distinct diagonal entries; the general case follows by a limiting argument).

Suppose that for all $t \in [0, 1]$ , the discs $D_{i_1}(t), \ldots, D_{i_m}(t)$ form a connected component $\Omega(t)$ separated from $D_{i_{m+1}}(t), \ldots, D_{i_n}(t)$ by a positive gap. By continuity, no eigenvalue path $\lambda_j(t)$ can jump from $\Omega(t)$ to the complement (or vice versa), since that would require passing through the gap where no disc exists. Therefore, the number of eigenvalues in $\Omega(t)$ is constant in $t$ . At $t = 0$ , exactly $m$ eigenvalues ( $a_{i_1}, \ldots, a_{i_m}$ ) lie in $\Omega(0)$ , so exactly $m$ eigenvalues lie in $\Omega(1)$ .

Remark on distinct diagonals. If some $a_{ii} = a_{jj}$ , perturb the diagonal entries by $\epsilon$ to make them distinct, apply the theorem, then take $\epsilon \to 0$ . The disc boundaries move continuously, preserving the count by a limiting argument. $\square$

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RemarkTightness and Extensions

Gershgorin's theorem is tight: for every configuration of discs satisfying the connectivity hypothesis, there exists a matrix realizing any permissible eigenvalue distribution. Extensions include the Brauer ovals of Cassini ( $|z - a_{ii}||z - a_{jj}| \leq R_i R_j$ for each pair $i,j$ ), which can give tighter localization, and the Brualdi regions based on directed graph structure of the matrix.

ExampleBlock Gershgorin Theorem

The Gershgorin theorem extends to block matrices: if $A = (A_{ij})$ with square diagonal blocks, then every eigenvalue lies in $\bigcup_i \{z : \sigma_{\min}(A_{ii} - zI) \leq \sum_{j \neq i}\|A_{ij}\|\}$ . For $2 \times 2$ blocks, this uses the smallest singular value instead of the simple modulus, giving tighter localization for matrices with natural block structure.