The generalized stability of families of real matrices and polynomials is considered. (Generalized stability is meant in the usual sense of confinementof matrix eigenvalues or polynomial zeros to a prespecified domain in the complex plane, and includes Hurwitz and Schur stability as special cases.)

"Guardian maps" and "semiguardian maps" are introduced as a unifying tool for the studyof this problem. Basically these are scalar maps that vanish when theirmatrix or polynomial argument loses stability. Such maps are exhibited for a wide variety of cases of interest corresponding to a generalized stability with respect to domains of the complex plane.

In the case of one- and two-parameter families of matrices or polynomials, concise necessary and sufficient conditions for generalized stability arederived. For the general multiparameter case, the problem is transformedinto one of checking that a given map is nonzero for the allowedparameter values.

Note: This is TR 88-69-r1. A previous version of this report, TR 88-69, had a different title.