Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form $$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$ where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks. Prove that $A^{-1}$ is similar to $\left(\begin{array}{cccc} J_{n_1}\left(\frac{1}{\lambda_1}\right) & 0 & \cdots & 0 \\ 0 & J_{n_2}\left(\frac{1}{\lambda_2}\right) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}\left(\frac{1}{\lambda_r}\right) \end{array}\right)$.
Let $A$ be a matrix of $\mathbf{GL}_n$ similar to its inverse. We admit that $A$ is similar to a block diagonal matrix of the form
$$A' = \left(\begin{array}{cccc} J_{n_1}(\lambda_1) & 0 & \ldots & 0 \\ 0 & J_{n_2}(\lambda_2) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \ldots & 0 & J_{n_r}(\lambda_r) \end{array}\right)$$
where the $\lambda_i$ are the eigenvalues of $A$ (not necessarily distinct) and $r$ as well as the $n_i$, $1 \leq i \leq r$, are nonzero natural integers. Moreover the matrix $A'$ is unique up to the order of the blocks.\\
Prove that $A^{-1}$ is similar to $\left(\begin{array}{cccc} J_{n_1}\left(\frac{1}{\lambda_1}\right) & 0 & \cdots & 0 \\ 0 & J_{n_2}\left(\frac{1}{\lambda_2}\right) & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & J_{n_r}\left(\frac{1}{\lambda_r}\right) \end{array}\right)$.