  The isospectral reduction of matrix, which is closely related to its Schur
complement, allows to reduce the size of a matrix while maintaining its
eigenvalues up to a known set. Here we generalize this procedure by increasing
the number of possible ways a matrix can be isospectrally reduced. The reduced
matrix has rational functions as entries. We show that the notion of
pseudospectrum can be extended to this class of matrices and that the
pseudospectrum of a matrix shrinks as the matrix is reduced. Hence the
eigenvalues of a reduced matrix are more robust to entry-wise perturbations
than the eigenvalues of the original matrix. We also introduce the notion of
inverse pseudospectrum (or pseudoresonances), which indicates how stable the
poles of a matrix with rational function entries are to certain matrix
perturbations. A mass spring system is used to illustrate and give a physical
interpretation to both pseudospectra and inverse pseudospectra.
