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This paper is essentially devoted to the study of the minimal eigenvalue \(\lambda _{N,\alpha }\) of the Toepllitz matrice \(T_N(\varphi _{\alpha })\) where \(\varphi _{\alpha }(e^{i \theta })=\vert 1- e^{i \theta } \vert ^{2\alpha } c_{1}(e^{i \theta })\) with \(c_{1}\) a positive sufficiently smooth function and \(0<\alpha <\frac{1}{2}\). We obtain \(\lambda _{N,\alpha }\sim c_{\alpha }N^{-2\alpha }c_{1}(1)\) when N goes to infinity and we obtain bounds for \(c_{\alpha }\). To arrive at these results we give a theorem which suggests that the entries of \(T_N^{-1}(\varphi _{\alpha })\) and \(T_N (\varphi ^{-1}_\alpha )\) are closely related. If \(\alpha _1 + \alpha _2 > \frac{1}{2}\) we obtain the asymptotic of the minimal eigenvalue of \(T_N (\varphi _{\alpha _1}) T_N (\varphi _{\alpha _2})\).

Valeur propre minimale d’une matrice de Toeplitz et d’un produit de matrices de Toeplitz