A generalization of Gershgorin circles (Q2805253)

The authors obtain results related to the Gershgorin circle theorem for a class of matrices. Recall that the Gershgorin circle theorem states that for an \(n \times n\) matrix \(A = (a_{ij})\), the spectrum of \(A\) is contained in the Gershgorin region NEWLINE\[NEWLINE G(A) = \bigcup_{i=1}^n \{z \in \mathbb{C} : |z - a_{ii}| \leq R_i\}, NEWLINE\]NEWLINE where NEWLINE\[NEWLINER_i = \sum_{j=1, j \neq i}^n |a_{ij}| \quad \quad \quad (i = 1, \dots, n).NEWLINE\]NEWLINE A sample result: Let \(A = (a_{ij})\) be an \(n \times n\) tridiagonal matrix and \(a_{i(i+1)} = a\), \(a_{(i+1)i} = b\) for all \(i\) and for some constants \(a\) and \(b\). Then, NEWLINE\[NEWLINE \sigma(A) \subseteq \bigcup_{i=2}^{n-1} \{z : |z - a_{ii}| \leq 2 \sqrt{|ab|}\} \cup \{z : |z - a_{11}| \leq |a|\} \cup \{z : |z - a_{nn}| \leq |b|\}. NEWLINE\]NEWLINE A sample application: Let \(p(t) = t^n + a_{n-1} t^{n-1} + \cdots + a_1 t + a_0\) be a monic polynomial. If \(|a_1| + \sqrt{|a_0|} \leq 1\), then the zeros of \(p(t)\) are contained in NEWLINE\[NEWLINE \bigcup_{i=3}^n \{z : |z| \leq 1 + |a_{i-1}|\}. NEWLINE\]