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Let $A$ be a positive definite $n \times n$ real matrix, $b \in \mathbb{R}^{n}$, and consider the quadratic polynomial
$Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle$
a) Show that $Q$ is bounded below, that is, there is a constant $m$ so that $Q(x) \geq m$ for all $x \in \mathbb{R}^{n}$.
b) Show that $Q$ blows up at infinity by showing that there are positive constants $R$ and $c$ so that if $\|x\| \geq R$, then $Q(x) \geq c\|x\|^{2}$.
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c) If $x_{0} \in \mathbb{R}^{n}$ minimizes $Q$, show that $A x_{0}=b$. [Moral: One way to solve $A x=b$ is to minimize $Q .$ ]
d) Give an example showing that if $A$ is only positive semi-definite, then $Q(x)$ may not be bounded below.
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