grandes-ecoles 2011 QV.B.3

grandes-ecoles · France · centrale-maths2__psi Invariant lines and eigenvalues and vectors Eigenvalue constraints from matrix properties

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.
Show that there exists a minimal real $k_0$ that we will specify as a function of the eigenvalues of $\Psi(M)$, such that the matrix $\Psi(N_{k_0})$ has non-negative eigenvalues.

We set $\xi_i^j = \begin{cases}0 & \text{if } i = j \\ 1 & \text{otherwise}\end{cases}$ and $N_k = (m_{ij} + k\xi_i^j)$ with $k$ a strictly positive real number.

Show that there exists a minimal real $k_0$ that we will specify as a function of the eigenvalues of $\Psi(M)$, such that the matrix $\Psi(N_{k_0})$ has non-negative eigenvalues.

Paper Questions

QI.A QI.B QI.C QII.A QII.B QIII.A QIII.B QIV.A.1 QIV.A.2 QIV.A.3 QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.B.5 QIV.B.6 QV.A.1 QV.A.2 QV.B.1 QV.B.2 QV.B.3 QV.C.1 QV.C.2 QV.C.3 QV.C.4 QV.C.5 QV.C.6 QV.C.7