grandes-ecoles 2013 QIV.B.3

grandes-ecoles · France · centrale-maths2__pc Invariant lines and eigenvalues and vectors Spectral properties of structured or special matrices

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.
What can be said about matrices whose eigenvalue circle is tangent to the $x$-axis and whose center is located on the $y$-axis?

In this section we consider a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and non-zero radius $r$, tangent to the $x$-axis. Let $A$ be a matrix whose eigenvalue circle equals $\mathcal{C}(\Omega, r)$.

What can be said about matrices whose eigenvalue circle is tangent to the $x$-axis and whose center is located on the $y$-axis?

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QI.C.1 QI.C.2 QI.C.3 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QIII.A.1 QIII.A.2 QIII.A.3 QIII.B.1 QIII.B.2 QIII.B.3 QIII.C QIII.D.1 QIII.D.2 QIII.D.3 QIII.E.1 QIII.E.2 QIII.E.3 QIII.E.4 QIV.A.1 QIV.A.2 QIV.A.3 QIV.A.4 QIV.A.5 QIV.A.6 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QV.A.1 QV.A.2 QV.B.1 QV.B.2 QV.C.1 QV.C.2