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e) We want to use corollary 55.7 in conjunction with part d) so that we have $A\cdot(x)/||A\cdot(x)||$ well defined and continuous on the non-negative part of $S^n$ and hence has a fixed point. Let $f(x) = A\cdot(e-x)/||A\cdot(e-x)||$ where $e \in R^{n+1}$ with all $1$ entries. $x \in B^{n+1}$ implies $||x||\leq 1$ and since $||e|| > 1$ at least one entry of $(e-x)$ is positive. Since $A$ is positive, $A\cdot (e-x)$ is positive so $f$ is well deifned. From d) we know $\exists x_0 \in B^{n+1}$ such that $f(x_0) = x_0$. Therfore, $A\cdot (e-x) = \lambda (e-x)$. Hence, lambda is positive eigen-value.
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Thu, 18 Mar 2021 19:13 GMT