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Gershgorin circle theorem

Contents

Context

Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

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 nn \in \mathbb{N} be a natural number.

Definition

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

  • the jjth Gershgorin radius of AA to be the sum of absolute values of off-diagonal entries in the jjth row:

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

    (2)D j A{z||za jj|r j A}. 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 AMat n×n()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. ):

cEV(A)1jn(cD j A). \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 kk, then it is contained in at least kk Gershgorin discs (Def. ).

This is due to Marsli-Hall 13.

Proposition

If the matrix has non-negavive 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)

References

The original article:

Textbook account:

Lecture notes:

Exposition:

  • Zack Cramer, The Gershgorin circle theorem 2017 (pdf)

See also:

Strengthening for eigenvalues with higher multiplicity:

Further strengthening for matrices with, in addition, non-negative entries:

  • Imre Bárány, József Solymosi, Gershgorin disks for multiple eigenvalues of non-negative matrices, In: M. Loebl, J. Nešetřil, R. Thomas (eds.) A Journey Through Discrete Mathematics, Springer, Cham. 2017 (arXiv:1609.07439, doi:10.1007/978-3-319-44479-6_6)

Last revised on April 27, 2021 at 09:58:55. See the history of this page for a list of all contributions to it.