grandes-ecoles 2025 Q22

grandes-ecoles · France · centrale-maths2__official Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Determine the eigenvalues of the matrix $A$. 
In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.

We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.

Determine the eigenvalues of the matrix $A$.
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