  In phylogenetics it is of interest for rate matrix sets to satisfy closure
under matrix multiplication as this makes finding the set of corresponding
transition matrices possible without having to compute matrix exponentials. It
is also advantageous to have a small number of free parameters as this, in
applications, will result in a reduction of computation time. We explore a
method of building a rate matrix set from a rooted tree structure by assigning
rates to internal tree nodes and states to the leaves, then defining the rate
of change between two states as the rate assigned to the most recent common
ancestor of those two states. We investigate the properties of these matrix
sets from both a linear algebra and a graph theory perspective and show that
any rate matrix set generated this way is closed under matrix multiplication.
The consequences of setting two rates assigned to internal tree nodes to be
equal are then considered. This methodology could be used to develop
parameterised models of amino acid substitution which have a small number of
parameters but convey biological meaning.
