Authors’ summary: Let be a set of complex matrices. For is the set of all products of matrices in of length . Denote by the joint spectral radius of , that is,
We call simultaneously contractible if there is an invertible matrix S such that
where is the spectral norm. This paper is primarily devoted to determining the optimal joint spectral radius range for simultaneous contractibility of bounded sets of complex matrices, that is, the maximum subset of such that if is a bounded set of complex matrices and , then is simultaneously contractible. The central result proved in this paper is that this maximum subset is Our method of proof is based on a matrix-theoretic version of complex John’s ellipsoid theorem and the generalized Gelfand spectral radius formula.