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\author{Harald R�ck}

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\textbf{First name:} Harald \\
\textbf{Last name:} R"ock \\
\textbf{Date:} \today \\
\textbf{Homework number:} 3 \\
\textbf{Homework Title:} Exercise 3.8 


\paragraph{Problem description:} ~\\
Supose that $\mathbf{A}$ is an $m \times n$ matrix of rank n. Prove that 
the matrix $\mathbf{A^T} \mathbf{A}$ is positive definite.

\paragraph{Problem solution:} ~\\
$\mathbf{A} = (a_{ij})_{i,j = 1}^{m,n}$ , 
$\mathbf{A^T} = (a'_{ij})_{i,j = 1}^{m,n} = (a_{ji})_{i,j = 1}^{n,m} $,
$x = (x_1, x_2 \dots , x_n)^T$ \\
We have to prove $\mathbf{x^T A^T A x} > 0$ :
\[ 
\Longrightarrow \mathbf{A^T A} = 
	\left( \sum_{k=1}^{n} a'_{ik}a_{kj} \right)_{i,j = 1}^{n,n} =
	\left( \sum_{k=1}^{n} a_{ki}a_{kj} \right)_{i,j = 1}^{n,n} =
	(c_{ij})_{i,j=1}^{n}
\]
\[ 
\Longrightarrow \mathbf{A^T A x} = 
	\left( \sum_{l=1}^{n} x_l c_{il} \right)_{i = 1}^{n} =
	\left( \sum_{l=1}^{n} x_l \sum_{k=1}^{n} a_{ki}a_{kl} \right)_{i = 1}^{n} =
	(x_i^*)_{i=1}^n
\]
\[ 
\Longrightarrow \mathbf{x^T A^T A x} = \sum_{h=1}^n x_h x_h^* =
	\sum_{h=1}^n x_h  \sum_{l=1}^{n} x_l \sum_{k=1}^{n} a_{kh}a_{kl} =
	\sum_{h=1}^n \sum_{l=1}^{n} \sum_{k=1}^{n} x_h x_l a_{kh}a_{kl} = \] 
\[ =
	\sum_{k=1}^n \sum_{l=1}^{n} \sum_{h=1}^{n} x_h x_l a_{kh}a_{kl} =
	\sum_{k=1}^n \sum_{l=1}^{n} x_l a_{kl} \sum_{h=1}^{n} x_h a_{kh}=
	\sum_{k=1}^n \left( \sum_{l=1}^{n} x_l a_{kl} \right)^2 
\]
\paragraph{Results}~\\
At the end we have a sum of positive numbers and at least one of them is 
not zero.
$\Longrightarrow \mathbf{x^T A^T A x} > 0$ 


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