A Jordan canonical form for reachable linear systems

AbstractWe construct a new canonical form for reachable matrix pairs (A, B) under the similarity action. The canonical pairs are explicitly characterized by the property that A is in Jordan canonical form and B is in a novel block echelon form whose block sizes are determined by the Jordan structure of A. The reachable pairs are classified by lists of indices, and it is shown that the canonical form is continuous on the Jordan strata consisting of all reachable pairs with the same index list. The Jordan strata are characterized topologically, and explicit dimension formulas are derived. Finally we indicate some open problems related to the Jordan decomposition of the space of reachable systems