grandes-ecoles 2025 Q19

grandes-ecoles · France · mines-ponts-maths1__psi Matrices Matrix Algebra and Product Properties

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set $$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$ For every $\lambda \in \mathbf{C}$ nonzero, $J_n(\lambda) = \lambda I_n + N$ where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise. Deduce from the previous questions that the matrix $J_n(1)$ is a product of two symmetry matrices.

For every polynomial $P = P(X) \in \mathbf{C}_{n-1}[X]$ we set
$$\left\{\begin{array}{l} s_1(P) = P(-X) \\ s_2(P) = P(1-X) \\ g(P) = P(X+1) - P(X) \end{array}\right.$$
For every $\lambda \in \mathbf{C}$ nonzero, $J_n(\lambda) = \lambda I_n + N$ where $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.\\
Deduce from the previous questions that the matrix $J_n(1)$ is a product of two symmetry matrices.

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