J Math Model Algor (2012) 11:119–157 DOI 10.1007/s10852-011-9170-4 Canonical Forms for Symmetric and Regular Structures Ali Kaveh · H. Fazli Received: 16 August 2011 / Accepted: 8 November 2011 / Published online: 3 December 2011 © Springer Science+Business Media B.V. 2011 Abstract Matrices associated with symmetric and regular structures can be arranged into certain block patterns known as Canonical forms. Using such forms, the de- composition of structural matrices into block diagonal forms, is considerably sim- plified. In this paper the main canonical forms are reviewed; and symmetric/regular structural configurations that can be explained with such forms are investigated. The invariant subspaces are formulated and the closed form solutions for the block- diagonalized stiffness matrices are provided in each case. Utility and robustness of the canonical forms in the analysis of structures exhibiting decomposable matrix patterns are demonstrated by numerous examples. Furthermore, a numerical method is proposed to extend the computational advantages of the matrix canonical forms to other nonconforming regular structures. Keywords Regular structure · Symmetric structure · Group theory · Matrix canonical form · Block diagonalization · Decomposition Mathematics Subject Classifications (2010) 15A18 · 74S05 1 Introduction Symmetric and regular structures commonly occur in engineering design because of the ease of construction, esthetic appeal and their optimal load-carrying capabilities. A structure is said to possess symmetry if through one or more symmetry operations its configuration becomes physically indistinguishable from the initial configuration. A. Kaveh (B ) · H. Fazli Centre of Excellence for Fundamental Studies in Structural Engineering, Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran-16, Iran e-mail: alikaveh@iust.ac.ir