Supporting vectors vs. principal components

Abstract

Let T : X -> Y be a bounded linear operator between Banach spaces X, Y. A vector x(0) is an element of S-X in the unit sphere S-X of X is called a supporting vector of T provided that parallel to T(x(0))parallel to = sup{parallel to T(x)parallel to : parallel to x parallel to = 1} = parallel to T parallel to. Since matrices induce linear operators between finite-dimensional Hilbert spaces, we can consider their supporting vectors. In this manuscript, we unveil the relationship between the principal components of a matrix and its supporting vectors. Applications of our results to real-life problems are provided.