New concepts in matrix theory (Q1239212)

Two new concepts introduced earlier by the author, the product transformation and the commutation matrix, are illustrated with particular examples. Any matrix \(M\) (of dimension \(N\)) can be expanded in the basis furnished by all possible products of all possible powers (a.p.p. of a.p.p.) of two matrices \(B\) and \(C\) (of dimension \(N\)) which satisfy the commutation \(CB=\omega BC\) where \(\omega\) is the primitive \(N^{th}\) root of unity. In this paper, it is demonstrated that a different basis in terms of a.p.p. of a.p.p. of matrices \({\mathcal L}_1,\dots,{\mathcal L}_{2n}, N=m^n\) obeying the ordered commutation relation \({\mathcal L}_i{\mathcal L}_j=\omega(m){\mathcal L}_j{\mathcal L}_i, i<j=1,2,\dots,2n,\) where \(\omega(m)\) is the \(m^{th}\) primitive root of unity can also be used for the expansion. The expansion coefficients are evaluated by transforming the \({\mathcal L}\) basis to the \({\mathcal L}'\) basis given by the the `helicity' matrices. The \({\mathcal L}\) and \({\mathcal L}'\) bases are related by a u-transformation which is explicitly carried out.