A Numerical Algorithm for Block-Diagonal Decomposition of Matrix ∗-Algebras, Part I: Proposed Approach and Application to Semidefinite Programming∗

Motivated by recent interest in group-symmetry in semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of matrices, or equivalently the irreducible decomposition of the generated matrix ∗-algebra. The method is composed of numerical-linear algebraic computations such as eigenvalue computation, and automatically makes full use of the underlying algebraic structure, which is often an outcome of physical or geometrical symmetry, sparsity, and structural or numerical degeneracy in the given matrices. The main issues of the proposed approach are presented in this paper under some assumptions, while the companion paper (Part II) gives an algorithm with full generality. Numerical examples of truss and frame designs are also presented.