grandes-ecoles 2022 Q8

grandes-ecoles · France · centrale-maths1__pc Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Show that $H$ belongs to $\mathcal{S}_n(\mathbb{R})$ and that its eigenvalues are strictly positive.

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Show that $H$ belongs to $\mathcal{S}_n(\mathbb{R})$ and that its eigenvalues are strictly positive.

Paper Questions

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