grandes-ecoles 2022 Q2

grandes-ecoles · France · centrale-maths1__official Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties

Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.
Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.

Let $M \in \mathrm { GL } _ { n } ( \mathbb { R } )$. Show that the eigenvalues of $M ^ { \top } M$ are all strictly positive.

Deduce that there exists a symmetric matrix $S$ with strictly positive eigenvalues such that $S ^ { 2 } = M ^ { \top } M$.

Paper Questions

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