Motivated by recent interest in group-symmetry in the area of semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of symmetric matrices, or equivalently the irreducible decomposition of the matrix ∗-algebra generated by symmetric matrices. The method does not require any algebraic structure to be known in advance, wh...

This paper proposes an approach for obtaining block diagonal and block triangular preconditioners that can be used for solving a linear system Ax = b, where A is a large, nonsingular, real, n × n sparse matrix. The proposed approach uses Tarjan’s algorithm for hierarchically decomposing a digraph into its strong subgraphs. To the best of our knowledge, this is the first work that uses this algo...

The capacitance matrix relates potentials and charges on a system of conductors. We review rigorously generalize its properties, block-diagonal structure inequalities, deduced from the geometry conductors analytic properties permittivity tensor. Furthermore, we discuss alternative choices regularization matrix, which allow us to find charge exchanged between having been brought an equal potenti...

Some engineering applications of heuristic multilevel optimization methods are presented and the discussion focuses on the dependency matrix that indicates the relationship between problem functions and variables. Decompositions are identified with dependency matrices that are full, block diagonal and block triangular with coupling variables. Coordination of the subproblem optimizations is show...

We describe a block matrix iterative algorithm for solving a linearquadratic parabolic optimal control problem (OCP) on a finite time interval. We derive a reduced symmetric indefinite linear system involving the control variables and auxiliary variables, and solve it using a preconditioned MINRES iteration, with a symmetric positive definite block diagonal preconditioner based on the parareal ...

In this paper, we show how to conjointly use module theory and constructive homological algebra to obtain general conditions for a matrix R of functional operators (e.g., differential/shift/time-delay operators) to be equivalent to a block-triangular or block-diagonal matrix R (i.e., conditions for the existence of unimodular matrices V and W satisfying that R = V R W ). These results allow us ...

A simple formula for the group inverse of a 2× 2 block matrix with a bipartite digraph is given in terms of the block matrices. This formula is used to give a graph-theoretic description of the group inverse of an irreducible tridiagonal matrix of odd order with zero diagonal (which is singular). Relations between the zero/nonzero structures of the group inverse and the Moore-Penrose inverse of...

We use the duality between a singly bordered and a doubly bordered block diagonal form to produce an efficient method of obtaining the stable factorization of a bordered system. This is particularly important in the exploitation of coarse-grained parallelism. We show how our partitioning and factorization scheme relates to domain decomposition partitioning but has greater control over stability...

In this paper, we present a systematic methodology for constructing LFT representations for general mechanical systems derived via Lagrange’s equations. The LFT representation allows for any nonlinear matrix second-order mechanical system to be transformed into an interconnection of an LTI system with a diagonal “uncertainty” block. This uncertainty block is, in fact, state-dependent. Sufficien...

In the present paper, we proposed a new efficient rank updating methodology for evaluating the rank (or equivalently the nullity) of a sequence of block diagonal Toeplitz matrices. The results are applied to a variation of the partial realization problem. Characteristically, this sequence of block matrices is a basis for the computation of the Weierstrass canonical form of a matrix pencil that ...