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A new method of matrix transformations. II. General theory of matrix diagonalizations via reduced characteristic equations and its application to angular momentum coupling

### Abstract

A general formalism is given to construct a transformation matrix which connects two matrices *A* and *B* of order *n*×*n* satisfying any given polynomialequation of degree *r*, *p* ^{(r)}(*x*) =0; *r*?*n*. The transformation matrix *T* _{ A B } 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 *T* _{ AΛ}, 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 λ_{ν} of *p* ^{(r)}(*x*), the transformation matrix *T* _{ A B } reduces to the idempotent matrix *P* _{ν} belonging to the eigenvalue λ_{ν} of *A*. Based on the relation which exists between *T* _{ AΛ} and *P* _{ν}, one can construct a transformation matrix *U* which is more effective than *T* _{ AΛ} and becomes unitary when *A* is Hermitian. Illustrative examples of the formalism are given for the problem of angular momentum coupling.

© 1979 American Institute of Physics

Published online 29 July 2008

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http://aip.metastore.ingenta.com/content/aip/journal/jmp/20/11/10.1063/1.523993

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/content/aip/journal/jmp/20/11/10.1063/1.523993

2008-07-29

2016-09-25

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