Weakly chained diagonally dominant matrix

In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.

of a complex matrix

is strictly diagonally dominant (SDD) if

{\displaystyle |a_{ii}|>\textstyle {\sum _{j\neq i}}|a_{ij}|}

Weakly diagonally dominant (WDD) is defined with

The directed graph associated with an

complex matrix

A complex square matrix

is said to be weakly chained diagonally dominant (WCDD) if The

A WCDD matrix is nonsingular.

[1] Proof:[2] Let

Suppose there exists a nonzero

in the null space of

Without loss of generality, let

is WCDD, we may pick a walk

ending at an SDD row

Taking moduli on both sides of and applying the triangle inequality yields and hence row

is WDD, the above chain of inequalities holds with equality so that

Repeating this argument with

is not SDD, a contradiction.

Recalling that an irreducible matrix is one whose associated directed graph is strongly connected, a trivial corollary of the above is that an irreducibly diagonally dominant matrix (i.e., an irreducible WDD matrix with at least one SDD row) is nonsingular.

[3] The following are equivalent:[4] In fact, WCDD L-matrices were studied (by James H. Bramble and B. E. Hubbard) as early as 1964 in a journal article[5] in which they appear under the alternate name of matrices of positive type.

WCDD L-matrix, we can bound its inverse as follows:[6] Note that

is always zero and that the right-hand side of the bound above is

Tighter bounds for the inverse of a WCDD L-matrix are known.

[7][8][9][10] Due to their relationship with M-matrices (see above), WCDD matrices appear often in practical applications.

WCDD L-matrices arise naturally from monotone approximation schemes for partial differential equations.

For example, consider the one-dimensional Poisson problem with Dirichlet boundary conditions

be a numerical grid (for some positive

that divides unity), a monotone finite difference scheme for the Poisson problem takes the form of and Note that

Venn Diagram showing the containment of weakly chained diagonally dominant (WCDD) matrices relative to weakly diagonally dominant (WDD) and strictly diagonally dominant (SDD) matrices.
The directed graph associated with the WCDD matrix in the example. The first row, which is SDD, is highlighted. Note that regardless of which node we start at, we can find a walk .