Let $B$ be the matrix in $M_{2}(\mathbf{R})$ defined by: $$B = \left(\begin{array}{cc} 3 & 2 \\ -5 & 1 \end{array}\right)$$ Prove that $B$ is semi-simple and deduce the existence of an invertible matrix $Q$ in $M_{2}(\mathbf{R})$ and two real numbers $a$ and $b$ to be determined such that: $$B = Q \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) Q^{-1}$$ Hint: for an eigenvector $V$ of $B$, one may introduce the vectors $W_{1} = \operatorname{Re}(V)$ and $W_{2} = \operatorname{Im}(V)$.
Let $B$ be the matrix in $M_{2}(\mathbf{R})$ defined by:
$$B = \left(\begin{array}{cc} 3 & 2 \\ -5 & 1 \end{array}\right)$$
Prove that $B$ is semi-simple and deduce the existence of an invertible matrix $Q$ in $M_{2}(\mathbf{R})$ and two real numbers $a$ and $b$ to be determined such that:
$$B = Q \left(\begin{array}{cc} a & b \\ -b & a \end{array}\right) Q^{-1}$$
Hint: for an eigenvector $V$ of $B$, one may introduce the vectors $W_{1} = \operatorname{Re}(V)$ and $W_{2} = \operatorname{Im}(V)$.