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Let $A \in \mathbb{R}^{m \times n}$ be of full rank. Consider $X_{k+1}=(2k-X_{k}A)X_{k}$, $X_0 = \alpha A^{T}$.

Let $E_k = I-X_kA$,

Deduce that if $||E_{0} ||_{2}<1$, then $lim_{k \rightarrow \infty}E_k=0$,

I'm thinking it's something like this $||E_0||_{2} = max_{v \not = 0} ||E_{o} v || / ||v|| < 1$, therefore

$max_{v \not = 0} ||E_{0} v || < ||v||$.

Then maybe use the fact that $E_k = E_0^{k-1}$.

But, I'm stuck here.

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So your proving contrapositive or something. –  simplicity Jan 14 '13 at 10:45
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