  Stochastic matrices and positive maps in matrix algebras proved to be very
important tools for analysing classical and quantum systems. In particular they
represent a natural set of transformations for classical and quantum states,
respectively. Here we introduce the notion of pseudo-stochastic matrices and
consider their semigroup property. Unlike stochastic matrices,
pseudo-stochastic matrices are permitted to have matrix elements which are
negative while respecting the requirement that the sum of the elements of each
column is one. They also allow for convex combinations, and carry a Lie group
structure which permits the introduction of Lie algebra generators. The quantum
analog of a pseudo-stochastic matrix exists and is called a pseudo-positive
map. They have the property of transforming a subset of quantum states
(characterized by maximal purity or minimal von Neumann entropy requirements)
into quantum states. Examples of qubit dynamics connected with "diamond" sets
of stochastic matrices and pseudo-positive maps are dealt with.
