Elementary Uniform Matrix Symmetrization

R. Purushothaman Nair

Advanced Technology Vehicles and Sounding Rockets Project(ATVP), Vikram Sarabhai Space Centre(VSSC), ISRO P.O, Thiruvananthapuram-695022, India. 

Abstract: This paper introduces a symmetrization process for a given matrix A∈R^n×n using elementary column(row) operations. Transformed symmetric matrix S∈R^n×n, S = (s_ij) has a structure S = (s_ij); i, j = 1 : n; s_ik = s_kj = s_kk; k = 1 : n for all i, j> k. This process is applicable to any matrix A∈R^n×n in a generalized way. Existing equivalence symmetrization of A in the literature is derived from it, providing identical result. Classical Cholesky factorization in the literature is revisited in the context of this symmetrization process. Elementary matrices apply equal scaling quantities with opposite signs in resultant matrices so that column(row) entries are identical with the corresponding diagonal entries. Because of this uniformity in scaling as well as matrix S, it may be called elementary uniform matrix symmetrization.

Keywords: matrix symmetrization, equivalence symmetrization, matrix factorization, linear systems.

Pages: 140 – 156 | Full PDF Paper