where Cm2 is an m2 x m2 tridiagonal block matrix with m x m matrices
/ 6 -1- dih
-1 + |d2 6 -1 hd2
(3.19)
on the diagonal block, and with negative m x m identity matrices on both its
super-diagonal and sub-diagonal blocks.
Lemma 3.2.6 Let M be any nx n tridiagonal matrix. If all entries on the diagonal,
superdiagonal, and subdiagonal of M are nonzero, then M is irreducible.
1 2 3 n-1 n
Figure 3-2: Directed graph, G(M), for a tridiagonal matrix M with nonzero entries
on its diagonal, superdiagonal, and subdiagonal.
Proof: Since the tridiagonal matrix M has nonzero entries on its diagonal,
superdiagonal, and subdiagonal, the directed graph of the matrix M, G(M), is
shown in Figure 3-2. From Figure 3-2, it is easy to see that G(M) is strongly
connected, and therefore M is irreducible by Theorem 3.2.3. *
Lemma 3.2.7 The matrix C in (3.18) is irreducible.
Proof: First, we want to show that each diagonal block Cm2 of the matrix C is
irreducible. By Lemma 3.2.6, each diagonal block (3.19) of Cm2 is irreducible;
hence, there exists a path for any two vertices associated with a diagonal block
(3.19). Suppose we are given vertices i < j in two distinct diagonal blocks; the
subdiagonal blocks of Cm2 provides a path between i and i + m and between i + m
and i + 2m etc. Eventually, we reach a vertex k in the block containing j. And