Infinite-time reachable abundance of linear discrete systems with Jordan system matrix

      In my blog article “Reachable abundance of linear discrete systems with Jordan system matrix”(http://blog.sciencenet.cn/blog-3343777-1085609.html) it is proven that the reachable abundance is only related to the first row, but not other rows, of the input matrix $B$ according to the upper Jordan matrix $A$ . Therefore, we have the following volume computing for the infinite-time reachable abundance of linear discrete systems with Jordan system matrix.

     It is assumed that the linear discrete systems $\varSigma(A,B)$ are with some repeated roots, the system matrices can be transformed as the upper Jordan matrices, that is, the matrices of the system models can be represented as

$A=\left[\begin{array}{ccccc} \lambda & 0 & 0 & \cdots & 0\\ 1 & \lambda & 0 & \cdots & 0\\ 0 & 1 & \lambda & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \lambda \end{array}\right],\quad B=\left[\begin{array}{c} b_{1}\\ b_{2}\\ b_{3}\\ \vdots\\ b_{n} \end{array}\right],\quad\Gamma=\left[\begin{array}{c} b_{1}\\ 0\\ 0\\ \vdots\\ 0 \end{array}\right],$

The the infinite-time reachable abundance of linear discrete systems with Jordan system matrix can be proven as

    $\textrm{Vol}(R_{r,\infty})=V_{n}\left(C_{n}\left([B,AB,...,A^{k}B,...,]\right)\right)$

              $=V_{n}\left(C_{n}\left([\Gamma,A\Gamma,...,A^{k}\Gamma,...,]\right)\right)$

              $=\textbfsymbol{\frac{\left|b_{1}^{n}\right|}{\left(1-\lambda\right)^{n}\left(1-\lambda^{2}\right)^{n(n-1)/2}}}$