Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.