## 每週問題 December 2, 2013

Let $A$ be an $n\times n$ matrix with characteristic polynomial $p(t)=(t-1)^n$. Show that $A$ is similar to its inverse.

$J=J_1\oplus\cdots\oplus J_k=\begin{bmatrix} J_1&&\\ &\ddots&\\ &&J_k \end{bmatrix}$

$\displaystyle J_i=\begin{bmatrix} 1&1& & &\\ &1&1 & &\\ & &\ddots&\ddots&\\ & & & 1& 1\\ & & & & 1 \end{bmatrix}$

\displaystyle\begin{aligned} J&=\begin{bmatrix} J_1&&\\ &\ddots&\\ &&J_k \end{bmatrix}=\begin{bmatrix} M_1&&\\ &\ddots&\\ &&M_k \end{bmatrix}\begin{bmatrix} J_1^{-1}&&\\ &\ddots&\\ &&J_k^{-1} \end{bmatrix}\begin{bmatrix} M_1^{-1}&\\ &\ddots&\\ &&M_k^{-1} \end{bmatrix}\\ &=(M_1\oplus\cdots\oplus M_k)J^{-1}(M_1\oplus\cdots\oplus M_k)^{-1}\\ &=MJ^{-1}M, \end{aligned}

\displaystyle\begin{aligned} (J_i^{-1})^k(J_i-I)^k&=(I-J_i^{-1})^k=(-1)^k(J_i^{-1}-I)^k\neq 0,~~0\le k\le n_i-1\\ (J_i^{-1})^{n_i}(J_i-I)^{n_i}&=(I-J_i^{-1})^{n_i}=(-1)^{n_i}(J_i^{-1}-I)^{n_i}=0. \end{aligned}

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