Given an arbitrary complex matrix ##IMG## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn1.gif] {$A$} and a generic matrix ##IMG## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn2.gif] {$X$} we find a canonical basis for the Kronecker part of the bi-Lagrangian subspace with respect to the corresponding Poisson brackets on the Lie algebra ##IMG## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn3.gif] {$\mathfrak{gl}_n(\mathbb C)$} , and also find a system of functions in bi-involution corresponding to this basis. In particular, for nilpotent matrices ##IMG## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn1.gif] {$A$} we prove that all nonzero functions obtained by applying the Mishchenko-Fomenko argument shift method to the coefficients of the characteristic polynomial form the Kronecker part of the complete system of functions in b...

中文翻译：

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矩阵代数上一对兼容泊松括号的规范基础

给定任意复杂矩阵## IMG ## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn1.gif] {$ A $}和通用矩阵## IMG＃ ＃[http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn2.gif] {$ X $}我们找到了关于双Lagrangian子空间的Kronecker部分的规范基础到李代数上的相应泊松括号## IMG ## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn3.gif] {$ \ mathfrak {gl} _n（ \ mathbb C）$}，并且还找到了与此基础相对应的双对合函数系统。特别是对于幂等矩阵## IMG ## [http://ej.iop.org/images/1064-5616/211/6/838/MSB_211_6_838ieqn1。