nLab Gershgorin circle theorem

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Contents

Context

Linear algebra

Idea
Statement
Related theorems
References

Idea

While the eigenvalues of a diagonal matrix are, of course, equal to its diagonal entries, Gershgorin’s circle theorem (Gershgorin 31, Prop. below) provides upper bounds (the Gershgorin radii, Def. below) for general square matrices over the complex numbers on how far, in the complex plane, the eigenvalues can be from the values of the diagonal entries.

Statement

Let $n \in \mathbb{N}$ be a natural number.

Definition

(Gershgorin discs)
Let $A \in Mat_{n \times n}(\mathbb{C})$ be a square matrix with complex entries. For $j \in \{1, \cdots, n\}$ define

the $j$ th Gershgorin radius of $A$ to be the sum of absolute values of off-diagonal entries in the $j$ th row:

(1) $r_j^A \;\coloneqq\; \underset{k \neq j}{\sum} \left\vert a_{j k} \right\vert$
the $j$ th Gershgorin disc to be the closed disk in the complex plane centered at the $j$ th diagonal entry with radius the above $j$ th Gershgorin radius:

(2) $D_j^A \;\coloneqq\; \Big\{ z \in \mathbb{C} \,\Big\vert\, \left\vert z - a_{j j} \right\vert \,\leq\, r_j^A \Big\} \,.$

Proposition

(Gershgorin disc theorem)
For $A \in Mat_{n \times n}(\mathbb{C})$ a square matrix with complex entries, each of its eigenvalues is contained in at least one Gershgorin disc (Def. ):

\underset{ c \in EV(A) }{\forall} \;\; \underset{ 1 \leq j \leq n }{\exists} \;\; \big( c \,\in\, D_j^A \,\subset\, \mathbb{C} \big) \,.

(see, e.g., Meckes 19, Thm. 7.1)

Proposition

In the context of Prop. , if an eigenvalue has mulitplicity $k$ , then it is contained in at least $k$ Gershgorin discs (Def. ).

This is due to Marsli-Hall 13.

Proposition

If the matrix has non-negative values and an eigenvalue has multiplicity greater than 1, then it is contained within this disc of half the radius inside one of the Gershgorin discs (2).

(Barany-Solymosi 16, Theorem 1)

Related theorems

Cauchy interlace theorem

References

The original article:

Semyon Aronovich Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Bulletin de l’Académie des Sciences de l’URSS. Classe des sciences mathématiques et na, 1931, Issue 6, Pages 749–754 (mathnet:eng/izv5235)

Textbook account:

Richard S. Varga, Gershgorin and His Circles, Springer-Verlag, Berlin, 2004 (doi:10.1007/978-3-642-17798-9)

Lecture notes:

Mark Meckes, Section 7.1 of: Lecture notes on matrix analysis, 2019 (pdf, pdf)

Exposition:

Zack Cramer, The Gershgorin circle theorem 2017 (pdf)