Diagonally dominant matrix

Diagonal dominant matrices describe in numerical mathematics to a class of square matrices with an additional condition on its diagonal elements. The single term is diagonally dominant mixed diagonally dominant in the literature and used strictly for times times for weakly diagonally dominant. The following two terms are explained in more detail.

  • 2.1 Definition
  • 2.2 Features

Strictly diagonally dominant matrix

Definition

A matrix is called strictly (also: strict or strong) diagonally dominant if the amount of its diagonal elements are each greater than the sum of the amounts of each remaining line entries, ie if for all

This criterion will be referred to as strong row sum criterion, and is not equivalent to the corresponding column sum of criterion, but by definition is equivalent to the sum of the columns of the transposed matrix criterion.

Applications

Complexes, strictly diagonally dominant matrices are regular because of the Gerschgorin circles, as well as the obtained from them by zeroing certain entries upper and lower triangular matrices. In some methods for solving systems of equations (e.g., Gauss-Seidel, Jacobi or SOR method ) has the diagonal dominance of the matrix system, in particular the latter property, a sufficient criterion for the convergence of the process.

Weak diagonally dominant matrices

Definition

A matrix is called weakly diagonally dominant, if the amount of its diagonal elements are each greater than or equal to the sum of the amounts of each remaining line entries, ie if for all

Properties

  • The amount of the weakly diagonally dominant matrices thus includes the set of strictly diagonally dominant matrices.
  • Real, symmetric, weakly diagonally dominant matrices with non-negative diagonal entries are positive semidefinite.

Irreducible diagonally dominant matrix

In the numerical analysis of partial differential equations also another term used for stability considerations:

A matrix is called irreducible diagonally dominant if it is irreducible and weakly diagonally dominant and for at least one inequality

Applies.

Main diagonal Gershgorin circle theorem Successive over-relaxation Symmetric matrix Josef Stoer
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