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. What information can be extracted concerning the 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.

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.