In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.
We say row of a complex matrix is strictly diagonally dominant (SDD) if . We say is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with instead.
The directed graph associated with an complex matrix is given by the vertices and edges defined as follows: there exists an edge from if and only if .
A complex square matrix is said to be weakly chained diagonally dominant (WCDD) if
The matrix
is WCDD.
A WCDD matrix is nonsingular. [1]
Proof: [2] Let be a WCDD matrix. Suppose there exists a nonzero in the null space of . Without loss of generality, let be such that for all . Since 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 not SDD. Moreover, since is WDD, the above chain of inequalities holds with equality so that whenever . Therefore, . Repeating this argument with , , etc., we find that 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.
Moreover, if is an 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 whenever one or more of the constants is one.
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. An example is given below.
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 . Letting 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 is a WCDD L-matrix.
In mathematics, the weakly chained diagonally dominant matrices are a family of nonsingular matrices that include the strictly diagonally dominant matrices.
We say row of a complex matrix is strictly diagonally dominant (SDD) if . We say is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with instead.
The directed graph associated with an complex matrix is given by the vertices and edges defined as follows: there exists an edge from if and only if .
A complex square matrix is said to be weakly chained diagonally dominant (WCDD) if
The matrix
is WCDD.
A WCDD matrix is nonsingular. [1]
Proof: [2] Let be a WCDD matrix. Suppose there exists a nonzero in the null space of . Without loss of generality, let be such that for all . Since 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 not SDD. Moreover, since is WDD, the above chain of inequalities holds with equality so that whenever . Therefore, . Repeating this argument with , , etc., we find that 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.
Moreover, if is an 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 whenever one or more of the constants is one.
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. An example is given below.
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 . Letting 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 is a WCDD L-matrix.