Diagonal dominant matrix

In numerical mathematics, diagonal-dominant matrices denote a class of square matrices with an additional condition on their main diagonal elements . The single term diagonally dominant is used inconsistently in the literature, sometimes for strictly diagonally dominant and sometimes for weakly diagonally dominant . Both terms are explained in more detail below.

Strictly diagonally dominant matrix

definition

A matrix is called strict (also: strict or strong) diagonally dominant if the amounts of its diagonal elements are greater than the sum of the amounts of the remaining respective line entries , i.e. h. if for all true ${\displaystyle (n\times n)}$ ${\displaystyle A=(a_{ij})}$ ${\displaystyle a_{ii}}$ ${\displaystyle a_{ij}}$ ${\displaystyle i\in \{1,\ldots ,n\}}$

${\displaystyle \sum _{j=1 \atop j\neq i}^{n}|a_{ij}|<|a_{ii}|}$.

This criterion is also known as the strong row sum criterion and is not equivalent to the corresponding column sum criterion, but by definition it is equivalent to the column sum criterion of the transposed matrix .

Applications

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

Weak diagonal dominant matrices

definition

A matrix is called weakly diagonally dominant if the amounts of its diagonal elements are greater than or equal to the sum of the amounts of the remaining respective line entries , i.e. h. if for all true ${\displaystyle n\times n}$ ${\displaystyle A=(a_{ij})}$ ${\displaystyle a_{ii}}$ ${\displaystyle a_{ij}}$ ${\displaystyle i\in \{1,\ldots ,n\}}$

${\displaystyle \sum _{j=1 \atop j\neq i}^{n}|a_{ij}|\leq |a_{ii}|}$.

properties

• The set of weakly diagonally dominant matrices thus includes the set of strictly diagonally dominant matrices.
• Real, symmetrical , weakly diagonally dominant matrices with nonnegative diagonal entries are positive semidefinite .

Irreducible diagonally dominant matrix

In the numerics of partial differential equations , another term is also used for stability considerations:

A matrix is called irreducible diagonally dominant if it is irreducible and weakly diagonally dominant and for at least one the inequality ${\displaystyle n\times n}$ ${\displaystyle A=(a_{ij})}$ ${\displaystyle i\in \{1,\ldots ,n\}}$

${\displaystyle \sum _{j=1 \atop j\neq i}^{n}|a_{ij}|<|a_{ii}|}$

applies.

Individual evidence

1. ↑ ^{a b} Christian Kanzow: Numerics of linear systems of equations. Direct and iterative procedures . Springer, Berlin et al. 2005, ISBN 3-540-20654-X , p. 142-143 .
2. ^ Christian Voigt, Jürgen Adamy: Collection of formulas for the matrix calculation . Oldenbourg, Munich et al. 2007, ISBN 978-3-486-58350-2 , pp. 81 .
3. ^ Hans Rudolf Schwarz, Norbert Köckler: Numerical Mathematics . Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-8348-9282-9 , pp. 39 (Electronic Resource).
4. ↑ Josef Stoer , Roland Bulirsch : Introduction to Numerical Analysis (= Texts in Applied Mathematics. Vol. 12). 3. Edition. Springer, New York NY et al. 2002, ISBN 0-387-95452-X , Theorem 8.2.6.
5. ↑ Josef Stoer , Roland Bulirsch : Introduction to Numerical Analysis (= Texts in Applied Mathematics. Vol. 12). 3. Edition. Springer, New York NY et al. 2002, ISBN 0-387-95452-X , Theorem 8.2.9.