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.

Preliminaries
We say row $$i$$ of a complex matrix $$A = (a_{ij})$$ is strictly diagonally dominant (SDD) if $$|a_{ii}|>\textstyle{\sum_{j\neq i}}|a_{ij}|$$. We say $$A$$ is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with $$\geq$$ instead.

The directed graph associated with an $$m \times m$$ complex matrix $$A = (a_{ij})$$ is given by the vertices $$\{1, \ldots, m\}$$ and edges defined as follows: there exists an edge from $$i \rightarrow j$$ if and only if $$a_{ij} \neq 0$$.

Definition
A complex square matrix $$A$$ is said to be weakly chained diagonally dominant (WCDD) if
 * $$A$$ is WDD and
 * for each row $$i_1$$ that is not SDD, there exists a walk $$i_1 \rightarrow i_2 \rightarrow \cdots \rightarrow i_k$$ in the directed graph of $$A$$ ending at an SDD row $$i_k$$.

Example
The $$m \times m$$ matrix
 * $$\begin{pmatrix}1\\

-1 & 1\\ & -1 & 1\\ & & \ddots & \ddots\\ & &  & -1 & 1 \end{pmatrix} $$ is WCDD.

Nonsingularity
A WCDD matrix is nonsingular.

Proof: Let $$A=(a_{ij})$$ be a WCDD matrix. Suppose there exists a nonzero $$x$$ in the null space of $$A$$. Without loss of generality, let $$i_1$$ be such that $$|x_{i_1}|=1\geq|x_j|$$ for all $$j$$. Since $$A$$ is WCDD, we may pick a walk $$i_1\rightarrow i_2\rightarrow\cdots\rightarrow i_k$$ ending at an SDD row $$i_k$$.

Taking moduli on both sides of
 * $$-a_{i_1 i_1}x_{i_1} = \sum_{j\neq i_1} a_{i_{1} j}x_j$$

and applying the triangle inequality yields
 * $$\left|a_{i_1 i_1}\right|\leq\sum_{j\neq i_1}\left|a_{i_1 j}\right|\left|x_j\right|\leq\sum_{j\neq i_1}\left|a_{i_1 j}\right|,$$

and hence row $$i_1$$ is not SDD. Moreover, since $$A$$ is WDD, the above chain of inequalities holds with equality so that $$|x_{j}|=1$$ whenever $$a_{i_1 j}\neq0$$. Therefore, $$|x_{i_2}|=1$$. Repeating this argument with $$i_2$$, $$i_3$$, etc., we find that $$i_k$$ is not SDD, a contradiction. $$\square$$

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.

Relationship with nonsingular M-matrices
The following are equivalent:
 * $$A$$ is a nonsingular WDD M-matrix.
 * $$A$$ is a nonsingular WDD L-matrix;
 * $$A$$ is a WCDD L-matrix;

In fact, WCDD L-matrices were studied (by James H. Bramble and B. E. Hubbard) as early as 1964 in a journal article in which they appear under the alternate name of matrices of positive type.

Moreover, if $$A$$ is an $$n\times n$$ WCDD L-matrix, we can bound its inverse as follows:
 * $$\left\Vert A^{-1}\right\Vert _{\infty}\leq\sum_{i}\left[a_{ii}\prod_{j=1}^{i}(1-u_{j})\right]^{-1}$$  where   $$u_{i}=\frac{1}{\left|a_{ii}\right|}\sum_{j=i+1}^{n}\left|a_{ij}\right|.$$

Note that $$u_n$$ is always zero and that the right-hand side of the bound above is $$\infty$$ whenever one or more of the constants $$u_i$$ is one.

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

Applications
Due to their relationship with M-matrices (see above), WCDD matrices appear often in practical applications. An example is given below.

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

For example, consider the one-dimensional Poisson problem
 * $$u^{\prime \prime}(x) + g(x)= 0$$  for   $$x \in (0,1)$$

with Dirichlet boundary conditions $$u(0)=u(1)=0$$. Letting $$\{0,h,2h,\ldots,1\}$$ be a numerical grid (for some positive $$h$$ that divides unity), a monotone finite difference scheme for the Poisson problem takes the form of
 * $$-\frac{1}{h^2}A\vec{u} + \vec{g} = 0$$  where   $$[\vec{g}]_j = g(jh)$$

and
 * $$A = \begin{pmatrix}2 & -1\\

-1 & 2 & -1\\ & -1 & 2 & -1\\ & & \ddots & \ddots & \ddots\\ & &  & -1 & 2 & -1\\ &  &  &  & -1 & 2 \end{pmatrix}.$$ Note that $$A$$ is a WCDD L-matrix.