Gershgorin circle theorem

In mathematics, the Gershgorin circle theorem may be used to bound the spectrum of a square matrix. It was first published by the Soviet mathematician Semyon Aronovich Gershgorin in 1931. The spelling of S. A. Gershgorin's name has been transliterated in several different ways, including Geršgorin, Gerschgorin, Gershgorin, Hershhorn, and Hirschhorn.

Let ${\displaystyle A}$ be a complex ${\displaystyle n\times n}$ matrix, with entries ${\displaystyle a_{ij}\,}$. For ${\displaystyle i\in \{1,\dots ,n\}}$ let ${\displaystyle R_{i}=\sum _{j\neq {i}}\left|a_{ij}\right|}$ be the sum of the absolute values of the non-diagonal entries in the ${\displaystyle i}$-th row. Let ${\displaystyle D(a_{ii},R_{i})}$ be the closed disc centered at ${\displaystyle a_{ii}}$ with radius ${\displaystyle R_{i}}$. Such a disc is called a Gershgorin disc.

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