grandes-ecoles 2015 QIV.B

grandes-ecoles · France · centrale-maths1__pc Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
Show that if $n$ is odd, then $f$ admits at least one real eigenvalue. 
In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.

Show that if $n$ is odd, then $f$ admits at least one real eigenvalue.
Paper Questions
QI.A QI.B QI.C QI.D QII.A QII.B QIII.A QIII.B QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QV.A QV.B QV.C QV.D