The Boolean pivot operation M-matrices, and reducible matrices

Boolean Matrices . . . Neither Boolean nor Matrices

Boolean matrices, the incidence matrices of a graph, are known not to be the (universal) matrices of a Boolean algebra. Here, we also show that their usual composition cannot make them the matrices of any algebra. Yet, later on, we “show” that it can. This seeming paradox comes from the hidden intrusion of a widespread set-theoretical (mis) definition and notation and denies its harmlessness. A...

Two modified Jacobi methods for M-matrices

E-Clean Matrices and Unit-Regular Matrices

Let $a, b, k,in K$ and $u, v in U(K)$. We show for any idempotent $ein K$, $(a 0|b 0)$ is e-clean iff $(a 0|u(vb + ka) 0)$ is e-clean and if $(a 0|b 0)$ is 0-clean, $(ua 0|u(vb + ka) 0)$ is too.

Multicolor urn models with reducible replacement matrices

Consider the multicolored urn model where after every draw, balls of the different colors are added to the urn in proportion determined by a given stochastic replacement matrix. We consider some special replacement matrices which are not irreducible. For three and four color urns, we derive the asymptotic behavior of linear combinations of number of balls. In particular, we show that certain li...

Matrices with Maximum Upper Multiexponents in the Class of Primitive, Nearly Reducible Matrices

B. Liu has recently obtained the maximum value for the kth upper multiexponents of primitive, nearly reducible matrices of order n with 1 ≤ k ≤ n. In this paper primitive, nearly reducible matrices whose kth upper multiexponents attain the maximum value are completely characterized.