grandes-ecoles 2016 QIV.C

grandes-ecoles · France · centrale-maths1__mp 3x3 Matrices Matrix Algebraic Properties and Abstract Reasoning

For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$. Let $\mu$ be the multiplicity of $r$ as an eigenvalue of $A$ and let $T = PAP^{-1}$ be a triangular reduction of $A$.
By examining the diagonal of $\left(\frac{1}{r}T\right)^m$ when $m \rightarrow +\infty$, show that $\mu = 1$.

For any primitive matrix $A$ in $\mathcal{M}_n(\mathbb{R})$, we denote $r$ the spectral radius of $A$. Let $\mu$ be the multiplicity of $r$ as an eigenvalue of $A$ and let $T = PAP^{-1}$ be a triangular reduction of $A$.

By examining the diagonal of $\left(\frac{1}{r}T\right)^m$ when $m \rightarrow +\infty$, show that $\mu = 1$.

Paper Questions

QI.A.1 QI.A.2 QI.B.1 QI.B.2 QII.A QII.B QII.C QIII.A QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.B.5 QIII.C.1 QIII.C.2 QIII.C.3 QIII.D.1 QIII.D.2 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C QV.A.1 QV.A.2 QV.A.3 QV.A.4 QV.A.5 QV.B.1 QV.B.2 QV.C.1 QV.C.2 QVI.A QVI.B.1 QVI.B.2 QVI.B.3 QVI.C.1 QVI.C.2 QVI.D