Maximum principles for elliptic systems and the problem of the minimum matrix norm of a characteristic matrix, revisited

In 1968, the existence of a maximum principle for some systems of partial differential equations led us to the following problem (see I. A. Rus, Studia Univ. Babes-Bolyai, 15(1968), No. 1, 19-26 and Glasnik Matematicki, 5(1970), No. 2, 356): Let A is an element of R-nxn be a matrix and parallel to.parallel to(2) the spectral norm on R-nxn. The problem is to determine, min(x is an element of R)parallel to A - xI parallel to(2). In this paper we study the evolution of this interesting relation between the theory of partial differential equations and the matrix theory. An application of an elliptic partial differential equation with complex valued coefficients is presented. New maximum principles are given and the case of infinite systems is also studied. Some open problems are formulated.