grandes-ecoles 2010 QI.D

grandes-ecoles · France · centrale-maths1__psi Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix
We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l } 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ \mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\ 0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \end{array} \right)$$
I.D.1) Deduce from the previous questions the spectrum of $C$. Specify the multiplicity order of the eigenvalues. I.D.2) Is the matrix $C$ diagonalizable over $\mathbb{C}$? over $\mathbb{R}$? If yes, indicate a diagonal matrix similar to $C$. 
We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$

$$C = \left( \begin{array} { l l l l l l l } 
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0
\end{array} \right)$$

I.D.1) Deduce from the previous questions the spectrum of $C$. Specify the multiplicity order of the eigenvalues.\\
I.D.2) Is the matrix $C$ diagonalizable over $\mathbb{C}$? over $\mathbb{R}$? If yes, indicate a diagonal matrix similar to $C$.
Paper Questions
QI.A QI.B QI.C QI.D QI.E QII.A QII.B QII.C QII.D QIII.A QIII.B QIII.C QIII.D