On the solution space of discrete time AR-representations over a finite time horizon

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The main purpose of this work is to determine the forward and backward solution space of a nonregular discrete time AR-representation i.e. A ( σ ) ξ ( k )=0, in a finite time horizon where A ( σ ) is a polynomial matrix and σ is the forward shift operator. The construction of the behavior is based on the structural invariants of the polynomial matrix that describes the AR-representation i.e. the finite and infinite elementary divisors and the right and left minimal indices of A ( σ ).