Let B be an irreducible reversible substochastic matrix and suppose that [PI] and [PI]' are positive

diagonal matrices such that [[PI].

Setting s = 1 in the above theorem gives us the class of

diagonal matrices.

The light pentad contains three

diagonal matrices ([[beta].

n] of invertible

diagonal matrices acts on the set of symmetric matrices by the rule X x A = XAX and we consider the graph G on n vertices [v.

m)]; according to this definition,

diagonal matrices have zero bandwidth, tridiagonal ones have bandwidth m = 1, etc.

This means that the solver we derive below is also applicable for semiseparable plus

diagonal matrices.

We experimented with additional problems from structural analysis [6], and circuit simulations (obtained from Sandia National Labs), but these had column space bases that were block

diagonal matrices, with small blocks; hence these problems were not interesting for our purposes.

A fractional splitting set for U, V, and W is a set of k-by-k

diagonal matrices [D.

where the

diagonal matrices D, D' are supposed to be quaternion-valued, nonsingular, but otherwise arbitrary matrices.