On the Possibility of Transforming a Matrix into an Orthogonal One

Abstract

An orthogonal matrix S∈R^n×n, which satisfies the relation S^TS=SS^T=I_n, has some interesting properties. For example, the rows and columns are orthogonal each other, and all eigenvalues exist on the unit circle with center at the origin in the complex plane. In addition, it is well known that a symmetric matrix is transformed to a diagonal one by the linear transformation with orthogonal matrices. In this paper, we discuss the existence of a matrix M which makes M^TAM an orthogonal matrix, where A∈R^n×n is an arbitrary real matrix. Consequently, the existence of such a matrix M is not necessarily assured if M is restricted to a square one. However, we can find such a matrix M that makes M^TAM an orthogonal one if we lift the above restriction on M.