We say row
i
of a complex matrix
A
=
(
a
i
j
)
is strictly diagonally dominant (SDD) if
|
a
i
i
|
>
∑
j
≠
i
|
a
i
j
|
. We say
A
is SDD if all of its rows are SDD. Weakly diagonally dominant (WDD) is defined with
≥
instead.
The directed graph associated with an
m
×
m
complex matrix
A
=
(
a
i
j
)
is given by the vertices
{
1
,
…
,
m
}
and edges defined as follows: there exists an edge from
i
→
j
if and only if
a
i
j
≠
0
.
A complex square matrix
A
is said to be weakly chained diagonally dominant (WCDD) if
A
is WDD and
for each row
i
of
A
, there exists a walk in the directed graph of
A
from
i
to an SDD row.
Note that if
i
is itself an SDD row, the trivial walk
i
→
i
satisfies the second requirement in the above definition.
The matrix
(
1
−
1
1
−
1
1
⋱
⋱
−
1
1
)
is WCDD.
A WCDD matrix is nonsingular.
Proof: Let
A
=
(
a
i
j
)
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
≥
|
x
j
|
for all
j
. Since
A
is WCDD, we may pick a walk
i
1
→
i
2
→
⋯
→
i
m
ending at an SDD row
i
m
.
Taking modulii on both sides of
−
a
i
1
i
1
x
i
1
=
∑
j
≠
i
1
a
i
1
j
x
j
and applying the triangle inequality yields
|
a
i
1
i
1
|
≤
∑
j
≠
i
1
|
a
i
1
j
|
|
x
j
|
≤
∑
j
≠
i
1
|
a
i
1
j
|
,
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
≠
0
. Therefore,
|
x
i
2
|
=
1
. Repeating this argument with
i
2
,
i
3
, etc., we find that
i
m
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.
The following are equivalent:
A
is a 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
×
n
WCDD L-matrix, we can bound its inverse as follows:
∥
A
−
1
∥
∞
≤
∑
i
[
a
i
i
∏
j
=
1
i
(
1
−
u
j
)
]
−
1
where
u
i
=
1
|
a
i
i
|
∑
j
=
i
+
1
n
|
a
i
j
|
.
Note that
u
n
is always zero and that the right-hand side of the bound above is
∞
whenever one or more of the constants
u
i
is one.
Tighter bounds for the inverse of a WCDD L-matrix are known.
Due to their relationship with M-matrices (see above), WCDD matrices appear often in practical applications. Some examples are given below.
Consider a Markov chain on the finite state space
{
1
,
…
,
m
}
. Let
{
π
}
be a (finite) set of stationary policies and
P
(
π
)
be the transition matrix for the chain under policy
π
. Let
ρ
(
π
)
∈
[
0
,
1
]
m
denote an
m
-vector corresponding to the discount factor at each state of the Markov chain. Because each
P
(
π
)
is a right stochastic matrix,
{
I
−
diag
(
ρ
(
π
)
)
P
(
π
)
}
is a family of WDD L-matrices. Convergence of R. A. Howard's policy iteration algorithm to the solution of a Markov decision process is guaranteed whenever this family consists only of inverse-positive matrices. As mentioned in a previous section, this occurs if and only if each member of the family is WCDD.
WCDD L-matrices arise naturally from monotone approximation schemes for partial differential equations.
For example, consider the one-dimensional Poisson problem
u
′
′
(
x
)
=
f
(
x
)
for
x
∈
(
0
,
1
)
with Dirichlet boundary conditions
u
(
0
)
=
u
(
1
)
=
0
. Letting
{
0
,
h
,
2
h
,
…
,
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
−
A
u
→
=
h
2
f
→
where
f
→
j
=
f
(
j
h
)
and
A
=
(
2
−
1
−
1
2
−
1
−
1
2
−
1
⋱
⋱
⋱
−
1
2
−
1
−
1
2
)
.
Note that
A
is a WCDD L-matrix.