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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.

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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.

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  • $\begingroup$ Okay, that is true for any orthogonal symmetric matrix. The question is concerned with finding some eigenvalues/eigenvectors of $JA$ which are made/obtained somehow by the eigenvalues/eigenvectors of $A$. I made some clarifications in the question. $\endgroup$
    – ABB
    Commented Jan 16, 2023 at 17:06
  • $\begingroup$ the eigenvalues and eigenvectors of $A$ fully determine the eigenvalues and eigenvectors of $JA$, but the algebraic relation for arbitrary $n$ is not simple, as you can see by considering small $n$. $\endgroup$
    – Carlo Beenakker
    Commented Jan 16, 2023 at 17:12
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    $\begingroup$ also, please avoid changing a question after it has been answered, or at least keep the original formulation for the record. $\endgroup$
    – Carlo Beenakker
    Commented Jan 16, 2023 at 17:17

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