Computing matrix symmetrizers, finally possible via the Huang and Nong algorithm

摘要

By a theorem of Frobenius (F.G. Frobenius, über die mit einer Matrix vertauschbaren Matrizen, Sitzungsberichte der K?niglich Preu?ischen Akademie der Wissenschaften zu Berlin (1910), pp. 3-15 (also in Gesammelte Abhandlungen, Band 3, Springer 1968. pp. 415-427)), every matrix A _(n,n) over any field F{double-struck} is the product of two symmetric ones. Using the algorithm of Huang and Nong (J. Huang and L. Nong, An iterative algorithm for solving finite-dimensional linear operator equations T(x) = f with applications, Linear Algebra Appl. 432 (2010), pp. 1176-1188) for linear systems, we develop an algorithm to compute a symmetric matrix S = S ~T ∈ F{double-struck}n,n for which SA is symmetric for any given square matrix A ∈ F{double-struck}n,n where F{double-struck} = ? or ?. The algorithm is implemented and tested in MATLAB.