Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the ***symmetric*** $2n$ by $2n$ matrix whose entries are given by, 

- $b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$. 
- $b_{n+k,l}=a_{kl}$ if $l$ is odd. 
- $b_{n+k,l}=-a_{kl}$ if $l$ is even.

Q. Let $V$ be a basis for the eigenvectors of $B$. Is there any approach to derive a basis for the eigenvectors of $A$ from $V$?