A cyclic six-term exact sequence for block matrices over a PID

Let A = ((A1)(0) (A2) (X)) be a 2 x 2 upper triangular block matrix over a principal ideal domain D with square diagonal blocks A(1) and A(2). We defi ne a cyclic six-term exact sequence epsilon (A) in terms of the kernels and the cokernels of A(1) and A(2) with a connecting map defined by the off di agonal block X. This cyclic sequence epsilon (A), under a variety of block- preserving matrix equivalences, is an invariant strictly finer than the Smi th normal forms of A(1), A(2) and A combined. As one example of how this ne w K-theoretic invariant is used in classical linear algebra, we prove that 2 x 2 upper triangular block matrices A and B over a field F are block-pres erving similar if and only if epsilon (tI - A) congruent to epsilon (tI - B ), that is, there is a chain F[t]-module isomorphism between the two cyclic sequences. We conclude with an application epsilon (A) to the flow equival ence classification of two-component shifts of finite type in symbolic dyna mics. This paper is self-contained and presented in a matrix-theoretic form .