CIRCUITS SYSTEMS SIGNAL PROCESS VOL. 8, NO. 1, 1989 ON THE REDUCIBILITYOF CENTROSYMMETRIC MATRICES APPLICATIONS IN ENGINEERING PROBLEMS* Lokesh Datta ~ and Salvatore D. Morgera 2 Abstract. The structure inherent to the centrosymmetric matrices is exploited to obtain factorization results leading to significant computational savings in many engineering applications. Several interesting properties of the matrices are discussed with a view toward algorithm computational complexity. It is shown that the multiplicative complexity involved in the process of principal component (eigen- value/eigenvector) extraction, and in the evaluation of the determinant and inverse of such matrices, can be reduced by nearly 75% by employing the results presented here. The theory presented here unifies and generalizes previous computationally efficient results on several specialized generalized centrosymmetric matrices; for example, the class of symmetric centrosymmetric (or doubly symmetric) matrices is shown to be a special case of the class of centrosymmetric matrices, and since the results obtained here are applicable over the field of complex numbers, the factorization results available for centrosymmetric matrices are readily extended to the complex field. The centrosymmetric matrices play an important role in a number of areas such as pattern recognition, antenna theory, mechanical and electrical systems, and quantum physics. Specific examples of pattern recognition feature selection, a uniform linear antenna array, vibration in structures, and the quantum- mechanical oscillator are discussed to demonstrate that the theory developed here has a wide range of applications. In addition, certain specialized cases of the centrosymmetric matrices have, in the past, proved their indisputable usefulness in the areas of communication theory, digital filters, linear systems, linear prediction, and speech analysis. 1. Introduction This work offers a number of interesting results on the class of centrosym- metric matrices. These matrices are often encountered in practice, and play * Received November 20, 1985; revised May 13, 1988. Research supported by NSERC Grant A0912. 1 Department of Electrical Systems Engineering, Wright State University, Dayton, Ohio 45435, USA. 2 Department of Electrical Engineering, McConnell Engineering Building, Room 514, McGill University, 3480 rue Universitr, Montrral H3A 2A7 Qurbec, Canada.