An Interpretable Orthogonal Decomposition of Positive Square Matrices

This study of square matrices with positive entries is motivated by a previous contribution on exchange rates matrices. The sample space of these matrices is endowed with a group operation, the componentwise product or Hadamard product. Also an inner product, identified with the ordinary inner product of the componentwise logarithm of the matrices, completes the sample space to be a Euclidean space. This situation allows to introduce two orthogonal decompositions: the first one inspired on the independence of probability tables, and the second related to the reciprocal symmetry matrices whose transpose is the componentwise inverse. The combination of them results in an orthogonal decomposition into easily computable four parts. The merit of this decomposition is that, applied to exchange rate matrices, the four matrices of the decomposition admit an intuitive interpretation.