grandes-ecoles 2015 QIV.D

grandes-ecoles · France · centrale-maths1__pc Invariant lines and eigenvalues and vectors Invariant subspaces and stable subspace analysis
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 $E$.
What do you think of the statement: ``every endomorphism of a finite-dimensional real vector space admits at least one line or one plane stable''? 
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 $E$.

What do you think of the statement: ``every endomorphism of a finite-dimensional real vector space admits at least one line or one plane stable''?
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