20
Definition 3.2.2 A directed graph is strongly connected if, for any ordered pair of
nodes Pi and Pj, there exists a directed path
PiPl, PIIP2 12 Pr-1 Pr=j
connecting Pi to Pj.
Theorem 3.2.3 An n x n matrix M is irreducible if and only if its directed graph
G(M) is strongly connected.
Definition 3.2.4 An n x n matrix M = (mij) is diagonally dominant if
n
Imi,i E Imij (3.16)
j=iljji
for all 1 < i < n. An n x n matrix M is irreducibly diagonally dominant zf M is
irreducible and diagonally dominant, with strict inequality in Inequality (3.16) for at
least one i.
Theorem 3.2.5 If M is an n x n irreducibly diagonally dominant matrix, then the
matrix M is nonsingular.
Now, we rearrange Equation (3.13) to yield
(A B) "+1= (AA + h(1 M)B n. (3.17)
Let C = A hMB and D = AA + h(1 ,.)B. Then we have
Cxn+l = DW",
where the matrix C is a block tridiagonal matrix given by
/ Cm m -I(m
-Im 2 Cm2 -Im2
C = (3.18)
h2