Abstract
It is proven, by purely algebraic techniques, that the matrixN, of (arbitrary) ordern, defined by\(N_{jk} = \delta _{jk} x_j \sum\limits_{l = 1}^n {\prime \left( {x_j - x_l } \right)^{ - 1} + \left( {1 - \delta _{jk} } \right)x_j \left( {x_j - x_k } \right)^{ - 1} } \) satisfies the Lax-type matrix equation.N= [M, N]. Here the dot indicates differentiation with respect tot andx i ≡ xi(t) arearbitrary functions oft. The explicit form of the matrixM is exhibited, as well as the left eigenvectors ofN (the eigenvalues ofN, that coincide with the firstn nonnegative integers, and its right eigenvectors were already known).
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