On Certain Linear Mappings Between Inner-Product and Squared-Distance Matrices

We obtain the spectral decomposition of four linear mappings. The first, Ie, is a mapping of the linear hull of all centered inner-product matrices onto the linear hull of all the induced squared-distance matrices. It is based on the natural generalization of the cosine law of elementary Euclidean geometry. The other three mappings studied are ,,1, the adjoint ,,*, and (" *)1. Extensions and applications, particularly to multidimensional scaling, are discussed in some detail.