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{\bf Pr{\'e}pa CAPES UPMC 2007-2008} \hfill Emmanuel FERRAND, Laurent KOELBLEN.
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\centerline{\bf TD d'Alg{\`e}bre du mercredi 16 janvier 2008}

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\section{} On consid{\`e}re les formes quadratiques
$q_1(x,y,z)=xy+yz+xz$,
$q_2(x,y,z,t)=xy+yz+zt+tx$,$q_3(x,y,z,t)=xy+yz+zt+tx+xy+yt$ . Pour
chacune d'entre elles, donner la matrice de la forme bilin{\'e}aire
sym{\'e}trique associ{\'e}e, son {\'e}ventuel noyau. Trouver une base orthogonale
pour $q_3$. 

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\section{} Soit $M$ une matrice carr{\'e}e symm{\'e}trique, soit
$\lambda_1,\dots,\lambda_n$ ses valeurs propres.
Montrer que $$\sum_i \lambda_i^2=\sum_{i,j}M_{i,j}^2.$$

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\section{D{\'e}composition $QR$}

Montrer que toute matrice carr{\'e}e r{\'e}elle $M$ se met sous la forme
$M=QR$, o{\`u} $Q$ est orthogonale et $R$ triangulaire sup{\'e}rieure.

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\section{In{\'e}galit{\'e} de Hadamard} Soit $A=(a_{i,j})_{i,j \in
  \{1,\dots,n\}}$ une matrice carr{\'e}e {\`a} coefficients r{\'e}els. 
Montrer que:

$$ |det(A)| \leq \prod_{j=1,\dots,n}( \sum_{i=1,\dots,n} a_{i,j}^2).$$

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\section{D{\'e}composition de Cholesky} Montrer que toute matrice r{\'e}elle
sym{\'e}trique $S$ se d{\'e}compose sous la forme $S={}^tLL$, o{\`u} $L$ est
triangulaire sup{\'e}rieure. Cette d{\'e}composition est-elle unique? qu'en
est-il si l'on impose que les coefficients diagonaux de $L$ sont tous positifs?


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\section{D{\'e}composition de Cholesky: exemple.}

Soit $S$ la matrice $n \times n$ dont les coefficients sont
$S_{i,j}=\binom{i-1}{j-1}$. Observer que $S$ est sym{\'e}trique et
trouver $L$ telle que $S={}^tLL$ (indication: commencer par les cas
$n=2$, $n=3$, ...).

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\section{D{\'e}composition Polaire}

Soit $S^+(n)$ l'ensemble des matrices sym{\'e}triques r{\'e}elles d{\'e}finies
positives $n \times n$. Soit $\Phi: O(n) \times S^+(n) \rightarrow
GL(n)$ l'application d{\'e}finie par $\Phi(Q,S)=QS$. Monter que $\Phi$ est
une bijection (c'est en fait un diff{\'e}omorphisme).
 

 
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\section{D{\'e}composition de Cayley}

\subsection{} Soit $A \in \mathcal M_n (\RM)$ une matrice antisym{\'e}trique. Montrer que
$(Id-A)$ est inversible et que $(Id+A)(Id-A)^{-1}$ est orthogonale.

\subsection{} Montrer toute matrice orthogonale sans valeur propre
{\'e}gale {\`a} $-1$ est de cette forme.

\subsection{} Quel est l'analogue de ce r{\'e}sultat pour les matrices
unitaires?

\subsection{} Dans le cas $O(2)$ et de $U(1)$, donner l'interpr{\'e}tation
g{\'e}om{\'e}trique des r{\'e}sultats pr{\'e}c{\'e}dents. 

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