Preliminaries

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

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

Definition

A complex square matrix ${\displaystyle A}$ is said to be weakly chained diagonally dominant (WCDD) if

• ${\displaystyle A}$ is WDD and
• for each row ${\displaystyle i_{1}}$ that is not SDD, there exists a walk ${\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}}$ in the directed graph of ${\displaystyle A}$ ending at an SDD row ${\displaystyle i_{k}}$.

Nonsingularity

A WCDD matrix is nonsingular.^[1]

Proof:^[2] Let ${\displaystyle A=(a_{ij})}$ be a WCDD matrix. Suppose there exists a nonzero ${\displaystyle x}$ in the null space of ${\displaystyle A}$. Without loss of generality, let ${\displaystyle i_{1}}$ be such that ${\displaystyle |x_{i_{1}}|=1\geq |x_{j}|}$ for all ${\displaystyle j}$. Since ${\displaystyle A}$ is WCDD, we may pick a walk ${\displaystyle i_{1}\rightarrow i_{2}\rightarrow \cdots \rightarrow i_{k}}$ ending at an SDD row ${\displaystyle i_{k}}$.

Taking moduli on both sides of

${\displaystyle -a_{i_{1}i_{1}}x_{i_{1}}=\sum _{j\neq i_{1}}a_{i_{1}j}x_{j}}$

and applying the triangle inequality yields

${\displaystyle \left|a_{i_{1}i_{1}}\right|\leq \sum _{j\neq i_{1}}\left|a_{i_{1}j}\right|\left|x_{j}\right|\leq \sum _{j\neq i_{1}}\left|a_{i_{1}j}\right|,}$

and hence row ${\displaystyle i_{1}}$ is not SDD. Moreover, since ${\displaystyle A}$ is WDD, the above chain of inequalities holds with equality so that ${\displaystyle |x_{j}|=1}$ whenever ${\displaystyle a_{i_{1}j}\neq 0}$. Therefore, ${\displaystyle |x_{i_{2}}|=1}$. Repeating this argument with ${\displaystyle i_{2}}$, ${\displaystyle i_{3}}$, etc., we find that ${\displaystyle i_{k}}$ is not SDD, a contradiction. ${\displaystyle \square }$

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.^[3]

Relationship with nonsingular M-matrices

The following are equivalent:^[4]

• ${\displaystyle A}$ is a nonsingular WDD M-matrix.
• ${\displaystyle A}$ is a nonsingular WDD L-matrix;
• ${\displaystyle A}$ 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^[5] in which they appear under the alternate name of matrices of positive type.

Moreover, if ${\displaystyle A}$ is an ${\displaystyle n\times n}$ WCDD L-matrix, we can bound its inverse as follows:^[6]

${\displaystyle \left\Vert A^{-1}\right\Vert _{\infty }\leq \sum _{i}\left[a_{ii}\prod _{j=1}^{i}(1-u_{j})\right]^{-1}}$   where   ${\displaystyle u_{i}={\frac {1}{\left|a_{ii}\right|}}\sum _{j=i+1}^{n}\left|a_{ij}\right|.}$

Note that ${\displaystyle u_{n}}$ is always zero and that the right-hand side of the bound above is ${\displaystyle \infty }$ whenever one or more of the constants ${\displaystyle u_{i}}$ is one.

Tighter bounds for the inverse of a WCDD L-matrix are known.^{[7][8][9][10]}

Monotone numerical schemes

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

For example, consider the one-dimensional Poisson problem

${\displaystyle u^{\prime \prime }(x)+g(x)=0}$   for   ${\displaystyle x\in (0,1)}$

with Dirichlet boundary conditions ${\displaystyle u(0)=u(1)=0}$. Letting ${\displaystyle \{0,h,2h,\ldots ,1\}}$ be a numerical grid (for some positive ${\displaystyle h}$ that divides unity), a monotone finite difference scheme for the Poisson problem takes the form of

${\displaystyle -{\frac {1}{h^{2}}}A{\vec {u}}+{\vec {g}}=0}$   where   ${\displaystyle [{\vec {g}}]_{j}=g(jh)}$

and

${\displaystyle A={\begin{pmatrix}2&-1\\-1&2&-1\\&-1&2&-1\\&&\ddots &\ddots &\ddots \\&&&-1&2&-1\\&&&&-1&2\end{pmatrix}}.}$

Note that ${\displaystyle A}$ is a WCDD L-matrix.