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Now, det [Ä(í) ] is nothing but a minor of det [GÀ1 ]. Thus, if we apply the Laplace theorem on expansion of a determinant to det [GÀ1 ] with respect to ní columns given by hÀ1 (í), changing í successively from 1 to r, we must have at least one set of nonvanishing minors such that r ‰ det [Ä(í) ] Tˆ 0 (2X4X11) íˆ1 with conditions hÀ1 (í) ’ hÀ1 (í9) ˆ 0, gÀ1 (í) ’ gÀ1 (í9) ˆ 0, for í Tˆ í9 (2X4X12) This means that there exists at least one sequence í ˆ g( j) of ëí for Ë that makes T non-singular for a given sequence í ˆ h(k).

An analogous treatment can be given for the Coulomb Dirac wave (Kim 1980c) and for the representations of the Lorentz group (Kim 1980a). 3 Intertwining matrices Let A be a matrix of order n 3 n that satis®es a polynomial equation of degree r (< n): P( r) (x) ˆ x r ‡ c1 x rÀ1 ‡ Á Á Á ‡ c r ˆ 0 (2X3X1) where c1 , c2 , F F F , c r are constant coef®cients. In terms of these coef®cients we de®ne a kth degree polynomial of x by x ( k) ˆ x k ‡ c1 x kÀ1 ‡ Á Á Á ‡ c k ; k ˆ 0, 1, 2, F F F , r (2X3X2a) with c0 ˆ 1, then the set satis®es the following recursion formulae: x ( k‡1) ˆ xx ( k) ‡ c k‡1 ; k ˆ 0, 1, F F F , r À 1 (2X3X2b) with x (0) ˆ 1.

Since A is real and symmetric, it can be diagonalized by a real orthogonal matrix. From A2 ˆ 4A, the reduced characteristic equation of A is given by A2 À 4A ˆ 0 which has no multiple roots. 3), we can immediately write down the transformation matrices T and T, P Q À3 1 1 1 T 1 À3 1 1U T U T ” T ˆ T ˆ A ‡ Ë À 41 ˆ T 1 1 À3 1U (2X7X3) U R S 1 1 1 1 One can easily see from this result that T is non-singular for any sequence of the characteristic roots in Ë. 3). They are symmetric since A is symmetric.

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