It is known from algebraic graph theory that if L the Laplacian matrix of some tree G with a vertex degree sequence d=(δ1,…,δn)⊤ and D its distance matrix, then LD+2I=(2⋅1−d)1⊤, where 1 an all-ones column vector. We prove converse proposition: this identity holds for d D, essentially tree, matrix. This result immediately generalizes to weighted graphs. Therefore, above bilinear equation in L, c...

We show that the Euclidean distance degree of a real orthogonally invariant matrix variety equals the Euclidean distance degree of its restriction to diagonal matrices. We illustrate how this result can greatly simplify calculations in concrete circumstances.

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

Multiblock analysis attacks the problem of how to combine data from various sources for purposes such as prediction, classification, clustering, or visual analysis. A key concept is distinction between “common” and “distinct” parts, that is, what information repeats itself across blocks unique an individual block. The statistical field multiblock holds many different approaches, which leads tre...

Biological genomes can be represented as square, symmetric, orthogonal, 0-1 matrices. It turns out that the rank distance applied to two genome matrices has a biological significance: it is related smallest number of basic rearrangement mutations, such reversals, translocations, transpositions (taken with weight 2), etc. explain differences between genomes. Therefore, closer will produce smalle...

In this paper we present diierent approaches to structur-ing covariance matrices within statistical classiiers. This is motivated by the fact that the use of full covariance matrices is infeasible in many applications. On the one hand, this is due to the high number of model parameters that have to be estimated, on the other hand the computational complexity of a classiier based on full covaria...