An algebraic tabulation is made of the Clebsch-Gordan (CG) coefficients of SU3 which occur in the reduction into irreducible representations of the direct product (λ, μ)⊗ (1, 1) of irreducible representations of SU3. Full explanation is made of the method of handling the complications associated with the possible double occurrence of the representation (λ, μ) itself in the direct product. The phase convention employed is an explicitly stated generalization of the well-known Condon and Shortley phase convention for SU2. The relationship of the CG coefficients associated with the direct product (1, 1)⊗ (λ, μ) to those coefficients already mentioned is also exhibited.
All the nonvanishing matrix elements of all the components of the tensor operator which belongs to the regular representation (the octet) of SU3 have been evaluated. Of special interest is the component 𝒴, for it is usual in the broken unitary symmetry theory of strong interactions to assume that the interactions which break exact SU3 invariance have the same transformation properties as 𝒴. Previously, matrix elements of 𝒴 connecting states of the same irreducible representation of SU3 have been given by Okubo in the form of the mass formula. Knowledge of all the matrix elements of 𝒴 is essential however if one is to do more than evaluate one-particle matrix elements in the broken unitary symmetry theory. Our method provides such knowledge for all components of the octet tensor operator with little more effort than is needed to treat 𝒴 alone.
Abstract. The least-squares method can be used to approximate an eigenvector for a matrix when only an approximation is known for the corresponding eigenvalue. In this paper, this technique is analyzed and error estimates are established proving that if the error in the eigenvalue is sufficiently small, then the error in the approximate eigenvector produced by the least-squares method is also small. Also reported are some empirical results based on using the algorithm.
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