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10.6 Bäcklund transformations 147To preserve the expansion of H(λ) as finite Taylor resp. Laurent series inλ we want to choose P (λ) such that H (λ) has the same pole structure asH 0 (λ). First we note that at λ = 0 and at λ = ∞ the poles of H (λ) are ofthe same order as the ones of H 0 (λ). It remains to analyse the behaviourof H (λ) at the zeros of det (P (λ)).Wehavem·n ∏det (P (λ)) = det (P m ) (λ − λ k ) , (10.72)where n is the size of the matrix. The zeros of the determinant, whichwe assume for the moment to be disjunct, are simple poles of the inversematrix, and withP (λ) −1 = ˜P (λ) / det (P (λ)) (10.73)we haveH (λ) = [det P (λ)] −1 [dP (λ)+P (λ) H 0 (λ)] ˜P (λ) . (10.74)Hence using (10.69) we requirek=1[dP (λ k )+P (λ k ) H 0 (λ k )] ˜P (λ k )= dΦ (λ k) ˜Φ(λ k )det (Φ 0 (λ k ))=0. (10.75)As d det(Φ 0 (λ)) = TrH (λ) = 0, we have without loss of generalitydet(Φ 0 (λ)) ̸= 0. Moreover, det (Φ (λ k )) = 0 and consequently there existnon-trivial eigenvectors p (k) such thatΦ(λ k ) p (k) =0,k=1,...,n· m. (10.76)Here, p (k) is a column vector of length n. For constant p (k) we also havedΦ (λ k ) p (k) = 0 and thusdΦ (λ k ) ˜Φ(λ k )=0. (10.77)This result follows from the fact that if two square matrices A and B havea common non-trivial eigenvector with eigenvalue zero, i.e. Ap = Bp =0,then A ˜B = 0. The matrices dΦ (λ k ) and ˜Φ(λ k ) are such a pair of matrices.The matrix P (λ) is to be constructed algebraically from the equationP (λ k )Φ 0 (λ k ) p (k) =0, k =1,...,n· m. (10.78)With a normalization condition, e.g. P m = 1, this is an inhomogeneousalgebraic system for the other expansion coefficients. Finally, H (λ) is calculatedvia (10.71). N.b., the constants λ k and the constant componentsof p (k) enter as parameters into the new solution. In a concrete case thechoice of parameters may be restricted by the requirement of preservingnot only the pole structure of H (λ) but also its ‘inner structure’, i.e. thealgebraic structure of its expansion coefficients. This may necessitate thatthe λ k and the components of p (k) are complex.