We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$ $$C = \left( \begin{array} { l l l l l l l }
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0
\end{array} \right)$$ Let $\Phi$ be the matrix in the basis $(f_1, f_2, f_3)$ of the endomorphism $\varphi$ of $F$ induced by $c$ (as determined in I.B). In this question, we propose to calculate the spectrum of $\Phi$ without calculating its characteristic polynomial. I.C.1) Why is 1 an eigenvalue of $\Phi$? I.C.2) Can we deduce from the sole calculation of the trace of $\Phi$ that $\Phi$ is diagonalizable in $\mathscr{M}_3(\mathbb{C})$? I.C.3) Calculate $\Phi^2$. Using the additional information obtained by calculating the trace of $\Phi^2$, determine the spectrum of $\Phi$. Is the matrix $\Phi$ diagonalizable in $\mathscr{M}_3(\mathbb{R})$?
We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l }
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0
\end{array} \right)$$
Let $\Phi$ be the matrix in the basis $(f_1, f_2, f_3)$ of the endomorphism $\varphi$ of $F$ induced by $c$ (as determined in I.B).
In this question, we propose to calculate the spectrum of $\Phi$ without calculating its characteristic polynomial.\\
I.C.1) Why is 1 an eigenvalue of $\Phi$?\\
I.C.2) Can we deduce from the sole calculation of the trace of $\Phi$ that $\Phi$ is diagonalizable in $\mathscr{M}_3(\mathbb{C})$?\\
I.C.3) Calculate $\Phi^2$. Using the additional information obtained by calculating the trace of $\Phi^2$, determine the spectrum of $\Phi$. Is the matrix $\Phi$ diagonalizable in $\mathscr{M}_3(\mathbb{R})$?