Prove that $\forall x , y \in \mathbb { R } ^ { n } , \left\langle A ^ { - 1 } x ; y \right\rangle = \left\langle x ; A ^ { - 1 } y \right\rangle$. Deduce that the matrix $A ^ { - 1 }$ is symmetric.
For $x , y \in \mathbb { R } ^ { n }$, we set: $( x ; y ) _ { A } = \langle A x ; y \rangle$. We denote by $E$ the endomorphism of the vector space $\mathbb { R } ^ { n }$ defined by $\forall x \in \mathbb { R } ^ { n } , E ( x ) = A ^ { - 1 } K x$. Prove that $( ; ) _ { A }$ defines an inner product on $\mathbb { R } ^ { n }$. Then show that $$\forall x , y \in \mathbb { R } ^ { n } , ( E ( x ) ; y ) _ { A } = ( x ; E ( y ) ) _ { A } .$$
Show that there exists a basis $\left( e _ { i } \right) _ { 1 \leq i \leq n }$ of $\mathbb { R } ^ { n }$ and $n$ strictly positive real numbers $\lambda _ { i } \in \mathbb { R } ^ { + * } ( 1 \leq i \leq n )$ such that $$\forall i \in \{ 1 , \ldots , n \} , A ^ { - 1 } K e _ { i } = \lambda _ { i } e _ { i }$$
We consider the differential equation: $$\forall t \in \left[ 0 , + \infty \left[ , A x ^ { \prime \prime } ( t ) = - K x ( t ) \right. \right. \tag{1}$$ with unknown function $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Show that $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ is a solution of the differential equation (1) if and only if there exist $2 n$ real numbers $\left( a _ { i } \right) _ { 1 \leq i \leq n } , \left( b _ { i } \right) _ { 1 \leq i \leq n }$ such that: $$\forall t \in \left[ 0 , + \infty \left[ , x ( t ) = \sum _ { i = 1 } ^ { n } \left( a _ { i } \cos \left( t \sqrt { \lambda _ { i } } \right) + b _ { i } \sin \left( t \sqrt { \lambda _ { i } } \right) \right) e _ { i } \right. \right.$$ Deduce that the set of solutions of (1) is a finite-dimensional vector space and specify its dimension.
Let $x , y \in C ^ { 1 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$. Prove that $$\forall t \in \left[ 0 , + \infty \left[ , \frac { d } { d t } ( \langle A x ; y \rangle ) ( t ) = \left\langle A x ^ { \prime } ( t ) ; y ( t ) \right\rangle + \left\langle A x ( t ) ; y ^ { \prime } ( t ) \right\rangle \right. \right.$$
Let $x \in C ^ { 2 } \left( \left[ 0 , + \infty \left[ ; \mathbb { R } ^ { n } \right) \right. \right.$ be a solution of the differential equation (1). For each real $t \geq 0$ we set, $T \left( x ^ { \prime } \right) ( t ) = \frac { 1 } { 2 } \left\langle A x ^ { \prime } ( t ) ; x ^ { \prime } ( t ) \right\rangle$ and $U ( x ) ( t ) = \frac { 1 } { 2 } \langle K x ( t ) ; x ( t ) \rangle$. Show then that the quantity $T \left( x ^ { \prime } \right) ( t ) + U ( x ) ( t )$ does not depend on $t \in [ 0 , + \infty [$.
Let $k \in \mathbb { N } ^ { * }$. We denote by $c _ { k }$ the positive real number such that: $c _ { k } \int _ { 0 } ^ { 2 \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } d t = 1$, and for every real $t$ we set $R _ { k } ( t ) = c _ { k } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k }$. Calculate $\int _ { 0 } ^ { \pi } \left( \frac { 1 + \cos t } { 2 } \right) ^ { k } \sin t \, d t$. Deduce that $c _ { k } \leq \frac { k + 1 } { 4 }$.