Prove that the application $$\begin{array} { | c l l }
\mathcal { M } _ { n } ( \mathbb { R } ) & \rightarrow & \mathbb { R } \\
M & \mapsto & \operatorname { tr } ( M )
\end{array}$$ is a linear form and that $$\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } , \quad \operatorname { tr } ( A B ) = \operatorname { tr } ( B A ) .$$
Show that the application $$\begin{array} { | c c c }
\left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } & \rightarrow & \mathbb { R } \\
( A , B ) & \mapsto & \operatorname { tr } \left( A ^ { \top } B \right)
\end{array}$$ is an inner product on $\mathcal { M } _ { n } ( \mathbb { R } )$.
Show that, if $A \in \mathcal { M } _ { n } ( \mathbb { R } )$ is nilpotent, then 0 is an eigenvalue of $A$ and that it is the only complex eigenvalue of $A$.
Suppose that $M , N$ and $M + N$ are nilpotent. By computing $( M + N ) ^ { 2 } - M ^ { 2 } - N ^ { 2 }$, show that $\operatorname { tr } ( M N ) = 0$.
Prove that a matrix $M$ in $\mathcal { M } _ { 2 } ( \mathbb { R } )$ is nilpotent if and only if $\operatorname { det } ( M ) = \operatorname { tr } ( M ) = 0$.
Suppose $n \geqslant 3$. Give an example of a matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ with zero trace and zero determinant, but not nilpotent.
Let $\left( E _ { 1 } , \ldots , E _ { n } \right)$ be the canonical basis of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. We denote $V = \sum _ { k = 1 } ^ { n } E _ { k }$. For $i \in \llbracket 1 , n \rrbracket$, express $E _ { i }$ in terms of $V$ and of $V - 2 E _ { i }$. Deduce that $\mathcal { M } _ { n , 1 } ( \mathbb { R } ) = \operatorname { Vect } \left( \mathcal { V } _ { n , 1 } \right)$.
Let $C _ { 1 } , \ldots , C _ { n }$ be $n$ column matrices in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$, with $C _ { 1 }$ non-zero. Prove that, if the family $\left( C _ { 1 } , \ldots , C _ { n } \right)$ is linearly dependent, then there exists a unique $j \in \llbracket 1 , n - 1 \rrbracket$ such that $$\left\{ \begin{array} { l }
\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent } \\
C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)
\end{array} \right.$$
Let $d \in \llbracket 1 , n \rrbracket , \left( U _ { 1 } , \ldots , U _ { d } \right)$ be a linearly independent family in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $H = \operatorname { Vect } \left( U _ { 1 } , \ldots , U _ { d } \right)$. Prove that there exist integers $i _ { 1 } , \ldots , i _ { d }$ satisfying $1 \leqslant i _ { 1 } < \cdots < i _ { d } \leqslant n$ such that the application $$\left\lvert \, \begin{array} { c c c }
H & \rightarrow & \mathcal { M } _ { d , 1 } ( \mathbb { R } ) \\
\left( \begin{array} { c }
x _ { 1 } \\
\vdots \\
x _ { n }
\end{array} \right) & \mapsto & \left( \begin{array} { c }
x _ { i _ { 1 } } \\
\vdots \\
x _ { i _ { d } }
\end{array} \right)
\end{array} \right.$$ is bijective. One may consider the rank of the matrix in $\mathcal { M } _ { n , d } ( \mathbb { R } )$ whose columns are $U _ { 1 } , \ldots , U _ { d }$.
Let $\mathcal { W }$ be a vector subspace of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$ of dimension $d$. Prove that $$\operatorname { card } \left( \mathcal { W } \cap \mathcal { V } _ { n , 1 } \right) \leqslant 2 ^ { d }$$
If $X$ follows the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$), specify the distribution of the random variable $\frac { 1 } { 2 } ( X + 1 )$.
Calculate the expectation and the variance of a variable following the distribution $\mathcal { R }$, where $\mathcal{R}$ is defined by $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$.
Let $X$ and $Y$ be two independent real random variables, each following the distribution $\mathcal { R }$ (where $X ( \Omega ) = \{ - 1,1 \}$, $\mathbb { P } ( X = - 1 ) = \mathbb { P } ( X = 1 ) = \frac { 1 } { 2 }$). Determine the distribution of their product $X Y$.
Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\tau _ { n } = \operatorname { tr } \left( M _ { n } \right)$. Calculate the expectation and the variance of the variable $\tau _ { n }$.
Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\delta _ { n } = \operatorname { det } \left( M _ { n } \right)$. Calculate the expectation of the variable $\delta _ { n }$.
Let $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ be $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$. The matrix random variable $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ takes values in $\mathcal { V } _ { n , n }$. We set $\delta _ { n } = \operatorname { det } \left( M _ { n } \right)$. Prove that the variance of the variable $\delta _ { n }$ is equal to $n!$ One may expand $\delta _ { n }$ along a row and reason by induction.
In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$. Calculate the probability of the event $M _ { 2 } \in \mathcal { N } _ { 2 }$.
In the particular case $n = 2$, $m _ { 11 } , m _ { 12 } , m _ { 21 }$ and $m _ { 22 }$ are four real random variables, mutually independent, all following the distribution $\mathcal { R }$ and $M _ { 2 } = \left( \begin{array} { l l } m _ { 11 } & m _ { 12 } \\ m _ { 21 } & m _ { 22 } \end{array} \right)$. Calculate the probability of the event $M _ { 2 } \in \mathcal { G } \ell _ { 2 } ( \mathbb { R } )$.
We consider $2n$ real random variables $c _ { 1 } , c _ { 2 } , \ldots , c _ { n }$ and $c _ { 1 } ^ { \prime } , c _ { 2 } ^ { \prime } , \ldots , c _ { n } ^ { \prime }$ that are mutually independent, all following the distribution $\mathcal { R }$. We consider the random column matrices $C = \left( \begin{array} { c } c _ { 1 } \\ \vdots \\ c _ { n } \end{array} \right)$ and $C ^ { \prime } = \left( \begin{array} { c } c _ { 1 } ^ { \prime } \\ \vdots \\ c _ { n } ^ { \prime } \end{array} \right)$. Prove that, for all $\omega \in \Omega$, the family $\left( C ( \omega ) , C ^ { \prime } ( \omega ) \right)$ is linearly dependent if and only if there exists $\varepsilon \in \{ - 1,1 \}$ such that $C ^ { \prime } ( \omega ) = \varepsilon C ( \omega )$.
We recall that $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ are $n ^ { 2 }$ real random variables that are mutually independent, all following the distribution $\mathcal { R }$, that $M _ { n } = \left( m _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n }$ is the random matrix taking values in $\mathcal { V } _ { n , n }$ and we denote $$C _ { 1 } = \left( \begin{array} { c }
m _ { 11 } \\
\vdots \\
m _ { n 1 }
\end{array} \right) , \ldots , C _ { n } = \left( \begin{array} { c }
m _ { 1 n } \\
\vdots \\
m _ { n n }
\end{array} \right)$$ the random variables taking values in $\mathcal { V } _ { n , 1 }$ constituted by the columns of the matrix $M _ { n }$. For all $j \in \llbracket 1 , n - 1 \rrbracket$, we denote by $R _ { j }$ the event $$\left( C _ { 1 } , \ldots , C _ { j } \right) \text { is linearly independent and } C _ { j + 1 } \in \operatorname { Vect } \left( C _ { 1 } , \ldots , C _ { j } \right)$$ and $R _ { n }$ the event $$\left( C _ { 1 } , \ldots , C _ { n } \right) \text { is linearly independent.}$$ Show that $( R _ { 1 } , \ldots , R _ { n } )$ is a complete system of events.
With the notation of question 28, deduce that $$\mathbb { P } \left( M \in \mathcal { G } \ell _ { n } ( \mathbb { R } ) \right) \geqslant \frac { 1 } { 2 ^ { n - 1 } } .$$
We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. Prove that the real number $$C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$$ exists and belongs to the interval $[ 0,1 ]$.
We denote by $I$ a subset of $\mathbb { N }$ having at least two elements and by $u = \left( u _ { i } \right) _ { i \in I }$ a sequence of unit vectors of $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. $C(u)$ denotes the coherence parameter $C ( u ) = \sup \left\{ \left| \left\langle u _ { i } \mid u _ { j } \right\rangle \right| , ( i , j ) \in I ^ { 2 } , i \neq j \right\}$. Show that if $C ( u ) = 0$, then the set $\left\{ u _ { i } , i \in I \right\}$ is finite and give an upper bound for its cardinality.
Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$ (where $X(\Omega) = \{-1,1\}$, $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=\frac{1}{2}$). We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. Prove that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) = \left( \operatorname { ch } \left( \frac { t } { n } \right) \right) ^ { n }$$
Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$. We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. Deduce that, for every real number $t$, $$\mathbb { E } ( \exp ( t \langle X \mid Y \rangle ) ) \leqslant \exp \left( \frac { t ^ { 2 } } { 2 n } \right)$$
Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$ By applying Markov's inequality to a suitably chosen random variable, prove that $$\forall t \in \mathbb { R } ^ { + } , \quad \mathbb { P } ( Z \geqslant \lambda ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } - \lambda t \right)$$
Let $\sigma$ and $\lambda$ be two strictly positive real numbers and $Z$ a real random variable such that $\exp ( t Z )$ has finite expectation and satisfies $$\forall t \in \mathbb { R } , \quad \mathbb { E } ( \exp ( t Z ) ) \leqslant \exp \left( \frac { \sigma ^ { 2 } t ^ { 2 } } { 2 } \right)$$ Deduce that $$\mathbb { P } ( | Z | \geqslant \lambda ) \leqslant 2 \exp \left( - \frac { \lambda ^ { 2 } } { 2 \sigma ^ { 2 } } \right)$$
Let $X _ { 1 } , \ldots , X _ { n } , Y _ { 1 } , \ldots , Y _ { n }$ be mutually independent random variables with the same distribution $\mathcal { R }$. We define the random vectors $X = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } , \ldots , X _ { n } \right) ^ { \top }$ and $Y = \frac { 1 } { \sqrt { n } } \left( Y _ { 1 } , \ldots , Y _ { n } \right) ^ { \top }$ taking values in $\mathcal { M } _ { n , 1 } ( \mathbb { R } )$. Prove that $$\mathbb { P } ( | \langle X \mid Y \rangle | \geqslant \varepsilon ) \leqslant 2 \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right)$$
$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$. Deduce from the previous questions that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) \leqslant N ( N - 1 ) \exp \left( - \frac { \varepsilon ^ { 2 } n } { 2 } \right) .$$
$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$. We assume that $n \geqslant 4 \frac { \ln N } { \varepsilon ^ { 2 } }$. Prove that $$\mathbb { P } \left( \bigcup _ { 1 \leqslant i < j \leqslant N } \left| \left\langle X ^ { i } \mid X ^ { j } \right\rangle \right| \geqslant \varepsilon \right) < 1 .$$
$N$ being a non-zero natural integer, $\left( X _ { j } ^ { i } \right) _ { 1 \leqslant i \leqslant N , 1 \leqslant j \leqslant n }$ is a family of $n \times N$ mutually independent real random variables with the same distribution $\mathcal { R }$. For every $i \in \llbracket 1 , N \rrbracket$, we set $X ^ { i } = \frac { 1 } { \sqrt { n } } \left( X _ { 1 } ^ { i } , \ldots , X _ { n } ^ { i } \right) ^ { \top }$. Deduce that, for every natural integer $N$ less than or equal to $\exp \left( \frac { \varepsilon ^ { 2 } n } { 4 } \right)$, there exists a family of $N$ unit vectors of $\mathbb { R } ^ { n }$ whose coherence parameter is bounded by $\varepsilon$.