On the algebraic characterization of a Mueller matrix in polarization optics. I. Identifying a Mueller matrix from its N matrix

Abstract

We revisit the problem of identifying a Mueller matrix M through N = [Mtilde]GM where G is the familiar Minkowski matrix diag (1, −1, −1, −1) and the tilde denotes matrix transposition. Using the standard methods of reduction of symmetric matrices (tensors) to their canonical forms in Minkowski space, we then show that there exist only two algebraically distinct types of Mueller matrices, which we call types I and II, and obtain the necessary and sufficient conditions for a Mueller matrix in terms of the eigenproperties of the associated N matrix. These conditions identify a Mueller matrix precisely and completely unlike the conditions derived earlier by Givens and Kostinski or by van der Mee. Observing that every Mueller matrix discussed hitherto in the literature is of the type I only, we construct examples of type-II Mueller matrices using the more familiar type-I (in fact pure Mueller) Mueller matrices. Further, we show that every G eigenvalue of an N matrix (see section 2 for a definition) is necessarily non-negative. Using this result, in an accompanying paper, we derive a general three-term factorization of a Mueller matrix which yields the general forms of Mueller and Jones-derived Mueller matrices and completely solves the problem of their algebraic structure.