grandes-ecoles 2023 Q5

grandes-ecoles · France · mines-ponts-maths1__psi Proof by induction Prove a general algebraic or analytic statement by induction

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. Show the inequality $\frac { \operatorname { Tr } ( M ) } { n } \geq \operatorname { det } ^ { 1 / n } ( M )$.
Hint: You may show that $x \mapsto - \ln ( x )$ is convex on $\mathbf { R } _ { + } ^ { \star }$.

Let $M \in S _ { n } ^ { + } ( \mathbf { R } )$ be a non-zero matrix. Show the inequality $\frac { \operatorname { Tr } ( M ) } { n } \geq \operatorname { det } ^ { 1 / n } ( M )$.

Hint: You may show that $x \mapsto - \ln ( x )$ is convex on $\mathbf { R } _ { + } ^ { \star }$.

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