grandes-ecoles 2019 Q5

grandes-ecoles · France · x-ens-maths__psi Matrices Eigenvalue and Characteristic Polynomial Analysis

Let $A \in \mathcal { S } _ { N } ( \mathbb { R } )$ and let $P \in \mathbb { R } [ X ]$ be a polynomial. Show that $P ( A ) \in \mathcal { S } _ { N } ( \mathbb { R } )$ and specify the eigenvalues and eigenvectors of $P ( A )$ in terms of those of $A$.

Let $A \in \mathcal { S } _ { N } ( \mathbb { R } )$ and let $P \in \mathbb { R } [ X ]$ be a polynomial. Show that $P ( A ) \in \mathcal { S } _ { N } ( \mathbb { R } )$ and specify the eigenvalues and eigenvectors of $P ( A )$ in terms of those of $A$.

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