A new method of matrix transformations. II. General theory of matrix diagonalizations via reduced characteristic equations and its application to angular momentum coupling
Shoon K. Kim · Journal of Mathematical Physics · 1979
A general formalism is given to construct a transformation matrix which connects two matrices A and B of order n×n satisfying any given polynomial equation of degree r, p(r)(x) =0; r?n. The transformation matrix TAB is explicitly given by a polynomial of degree (r−1) in A and B based on p(r)(x). A special case where B is a diagonal matrix Λ equivalent to A leads to the general theory of matrix diagonalizations with the transformation matrix TAΛ, which can be made nonsingular with a proper choice of Λ. In another special case where B is a constant matrix with the constant being a simple root λν of p(r)(x), the transformation matrix TAB reduces to the idempotent matrix Pν belonging to the eigenvalue λν of A. Based on the relation which exists between TAΛ and Pν, one can construct a transformation matrix U which is more effective than TAΛ and becomes unitary when A is Hermitian. Illustrative examples of the formalism are given for the problem of angular momentum coupling.