The aim of this work is to give a comprehensive overview of the problem of jointly block-diagonalizing a set of matrices. We discuss how to implement methods in the common case of only approximative block-diagonalizability using Jacobi algorithms. Standard Jacobi optimization techniques for diagonalization and joint diagonalization are reviewed first, before we study their generalizations to th...

In this short note we deal with a constructive scheme to decompose a continuous family of matrices A(ρ) asymptotically as ρ → 0 into blocks corresponding to groups of eigenvalues of the limit matrix A(0). We also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on Hilbert spaces. 1. Matrix theory 1.1. Preliminaries. We first recall some w...

A novel block diagonalisation (BD)-based Multiuser multiple input multiple output (MU-MIMO)-aided relaying scheme is proposed for downlink transmissions, which does not require any channel state information at the base station (BS) and decomposes a MU-MIMO-aided relaying system into several parallel single-user MIMO-assisted relaying schemes. Furthermore, based on the proposed algorithm the opt...

We consider the formal reduction of a system of linear differential equations and show that, if the system can be block-diagonalised through transformation with a ramified Shearing-transformation and following application of the Splitting Lemma [9], and if the spectra of the leading block matrices of the ramified system satisfy a symmetry condition, this block-diagonalisation can also be achiev...