2 The hypotheses are precised.

# A question about matrices withmoredetails

Let $A,B\in\mathbb{R}^{n\times n}$. Suppose that $B$ is nonsingular and that there exists $m$ reals pairwise distinct $\lambda_{1},\cdots,\lambda_{r},\cdots,\lambda_{m}$ such that $$B^{-1}A=\begin{pmatrix} \lambda_{1}& 1&&&&&&\cr &\lambda_{1}&\ddots&&&&&\cr &&\ddots&&&\LARGE{0}&&\cr &&&\lambda_{r}&1&&&\cr &&&&\lambda_{r}&&&\cr &&&&&\lambda_{r+1}&&\cr &&\LARGE{0}&&&&\ddots&\cr &&&&&&&\lambda_{m} \end{pmatrix}$$ (It is the canonical Jordan form) Can we always ever find reals numbers $t_{1},\cdots,t_{p}$ such t_ {1}, \cdots, t_ {p} $so that the two following assertions are true: 1. $$A\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B=B\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)A?$$A\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B=B\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)A$
2. $\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B\quad\mbox{is nonsingular and diagonalizable }$?

N.B :

1. The integer $p$ is not fixed.
2. This question has arisen when studying the contollability of a real discrete-time nonlinear system. This explains why the matrices are supposed to be reals.

Thanks for help.

1

Let $A,B\in\mathbb{R}^{n\times n}$. Can we always find reals $t_{1},\cdots,t_{p}$ such that $$A\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)B=B\Big(\displaystyle\prod_{i=1}^{p}(A+t_{i}B)\Big)A?$$