Let $A$ be a real symmetric matrix in $M\_{2n}(\mathbb{R})$with $A^2=I\_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$J=I\_n\oplus (-I\_n)$$


Q. By the eigen values/eigenvectors of $A$, can we find/make some eigenvalues/eigenvectors of the product $JA$?


p.s. We denote $I\_n$ by the $n\times n$ identity matrix and $-I\_n$ is its negative.



The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
**Original question:** *What information can be extracted concerning the eigenvalues/eigenvectors of the product $JA$?*


The matrix $JA$ is orthogonal,
$$JA(JA)^\top=JA^2J=J^2=I,$$
so its eigenvalues $\lambda\_p$ are complex conjugate pairs $e^{\pm i\phi\_p}$ on the unit circle. The eigenvectors are an orthonormal set.


