Sign eigenanalysis and its applications to optimization problems and robust statistics

Sign eigenvectors for a real square matrix, A, are defined to be sign vectors for which all of its elements either retain the same signs or become to their opposite signs after the linear transformation A, where a sign vector is a vector with the elements equal to either 1 or -1. Existence of sign eigenvectors for symmetric positive semi-definite matrices is investigated. It is shown that the sign eigenanalysis is closely related to some certain optimization problems and can be applied to develop robust statistical inference procedures in the L1 norm. A numerical example is given to illustrate the applications to robust multivariate statistical analysis.