grandes-ecoles 2016 QI.B.1

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

We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. We want to show that $\lim_{m \rightarrow +\infty} A^m = 0$.
Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.
Show that $\widehat{T}$ is upper triangular and that we can choose $\delta$ so that $N(\widehat{T}) < 1$.

We are given $A$ in $\mathcal{M}_n(\mathbb{C})$, with $\rho(A) < 1$. We want to show that $\lim_{m \rightarrow +\infty} A^m = 0$.

Let $P$ be in $\mathrm{GL}_n(\mathbb{C})$ and let $T$ be upper triangular, such that $A = PTP^{-1}$. We are given $\delta > 0$. We set $\Delta = \operatorname{diag}\left(1, \delta, \ldots, \delta^{n-1}\right)$ and $\widehat{T} = \Delta^{-1}T\Delta$.

Show that $\widehat{T}$ is upper triangular and that we can choose $\delta$ so that $N(\widehat{T}) < 1$.

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