Prove that the matrices $H_\phi^{(n)}$ and $H_\psi^{(n)}$ are similar.

Question

For any map $f: \mathbb{N}\to \mathbb{R}$ and any strictly positive integer $n$, We denote by $H_f^{(n)}$ the matrix of $\mathcal{M}_n$ given by: $$ H_f^{(n)}=(f(i + j -2))_{1\leq i,j\leq n} = $$ Let $n\geq 3$ and $\phi$ and $\psi$ are the functions defined on $\mathbb{N}$ by: $$ \forall k\in \mathbb{N},\qquad \phi (k) = (- 1)^ k (k + 1) \qquad,\qquad \psi (k) =k + 1 $$

Problem

Prove that the matrices $H_\phi^{(n)}$ and $H_\psi^{(n)}$ are similar.

An idea please

Answer 1

Each entry of $H_\phi^{(n)}$ has the same modulus as its counterpart in $H_\psi^{(n)}$. The only difference between the two matrices is a checkerboard sign pattern. Therefore $H_\phi^{(n)}=DH_\psi^{(n)}D^{-1}$ where $D=D^{-1}=\operatorname{diag}(1,-1,1,-1,\ldots,(-1)^{n-1})$, and the two matrices are similar.