We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$. Let $A$ be a matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$. We assume that its characteristic polynomial equals $X ^ { 2 } + r ^ { 2 }$, where $r$ is a non-zero real number. a) Justify the existence of a matrix $P \in GL ( 2 , \mathbb { C } )$ satisfying: $ir H _ { 0 } = P ^ { - 1 } A P$. What is the value of the matrix $A ^ { 2 } + r ^ { 2 } I _ { 2 }$? b) Let $f$ be the endomorphism of $\mathbb { R } ^ { 2 }$ canonically associated with the matrix $A$, that is, which maps a column vector $u$ of $\mathbb { R } ^ { 2 }$ to the vector $A u$. Let $w$ be a non-zero vector of $\mathbb { R } ^ { 2 }$. Prove that the family $\left( \frac { 1 } { r } f ( w ) , w \right)$ is a basis of $\mathbb { R } ^ { 2 }$, and give the matrix of $f$ in this basis.
We assume in this question that $\mathbb { K }$ is equal to $\mathbb { R }$.
Let $A$ be a matrix of $\mathcal { M } _ { 0 } ( 2 , \mathbb { R } )$. We assume that its characteristic polynomial equals $X ^ { 2 } + r ^ { 2 }$, where $r$ is a non-zero real number.
a) Justify the existence of a matrix $P \in GL ( 2 , \mathbb { C } )$ satisfying: $ir H _ { 0 } = P ^ { - 1 } A P$. What is the value of the matrix $A ^ { 2 } + r ^ { 2 } I _ { 2 }$?
b) Let $f$ be the endomorphism of $\mathbb { R } ^ { 2 }$ canonically associated with the matrix $A$, that is, which maps a column vector $u$ of $\mathbb { R } ^ { 2 }$ to the vector $A u$. Let $w$ be a non-zero vector of $\mathbb { R } ^ { 2 }$. Prove that the family $\left( \frac { 1 } { r } f ( w ) , w \right)$ is a basis of $\mathbb { R } ^ { 2 }$, and give the matrix of $f$ in this basis.