I was just thinking about if there exists any way to antidiagonalizate a matrix. I mean given A, finding a antidiagonal matrix J such that
$\pmatrix{v_1 \\ v_2 \\ \vdots\\ v_n} = \pmatrix{0 &\dots& 0&\lambda_{1,n} \\ 0 & \dots & \lambda_{2,n-1} & 0 \\ \vdots && &\vdots& \\ \lambda_{n,1} & \dots & 0 & 0}$ $\pmatrix{v_1 \\ v_2 \\ \vdots\\ v_n}$.
The only thing I thought was that if $\lambda_{n,k}$ is an antieigenvalue of A then $\lambda_{n,k}\cdot\lambda_{k,n}$ is an eigenvalue of $A^2$.
Any references? Thanks!