  There has recently been renewed recognition of the need to understand the
consistency properties that must be preserved when a generalized matrix inverse
is required. The most widely known generalized inverse, the Moore-Penrose
pseudoinverse, provides consistency with respect to orthonormal transformations
(e.g., rotations of a coordinate frame), and a recently derived inverse
provides consistency with respect to diagonal transformations (e.g., a change
of units on state variables). Another well-known and theoretically important
generalized inverse is the Drazin inverse, which preserves consistency with
respect to similarity transformations. In this paper we note a limitation of
the Drazin inverse is that it does not generally preserve the rank of the
linear system of interest. We then introduce an alternative generalized inverse
that both preserves rank and provides consistency with respect to similarity
transformations. Lastly we provide an example and discuss experiments which
suggest the need for algorithms with improved numerical stability.
