Spectral problem of block-rectangular hierarchical matrices
Block-hierarchical matrices are a special kind of structures where hierarchically the nested growing blocks are placed along a diagonal and each (sub-)block is again a block-hierarchical matrix itself. They were introduced for considerations in phase transitions, used for the theory of spin-glasses, multifractals, hierarchical networks, and complex systems, where such matrices describe the probabilities of transitions between different states of the system identified with the points of the ultrametric space. In this paper, the authors introduce a new class of non-symmetric block-hierarchical matrices. Each such \(p^r\times p^r\) matrix is composed of \(p\) identical \(p^r\times p^{r-1}\) rectangular sub-matrices which have the structure similar to conventional block-hierarchical matrices. Further, they describe their spectral properties and give an application: a simple error generating model that can be used in the theory of information. The paper is divided into five sections. The first one shows how the new type of matrices can be generated. The second one presents the tensor product representation of such matrices and the solution of the spectral problem for block-hierarchical matrices with connection to the tensor product. Section 3 presents the block-rectangular hierarchical matrices with their definition and spectral properties. In the next section we have a schematic model of error generation in information sequences as the one of possible applications of proposed matrices. The paper is concluded in Section 5.
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