"Induction sets and p-eigenvectors"Lócsi, LeventeThe well-known way to define the $p$-norm of a matrix is to find $\left\|A\right\|_p = \sup_{x \neq 0} \frac{\left\|Ax\right\|_p}{\left\|x\right\|_p}.$ The fraction present in this formula will be investigated more thoroughly: diving into some details usually overlooked in related discussions. Induction sets (induction curves or induction surfaces in low dimensions) provide a visual aid to examine these fractions. Note that eigenvectors of the matrix naturally have the property that the fraction at hand is independent of $p \in [1,\infty]$. We will show that there exist further non-trivial vectors for a given matrix with this peculiar property. These are to be called $p$-eigenvectors. Analytical results and construction methods shall be presented alongside nice illustrations, handy Matlab scripts and some open problems related to higher dimensional operators. |
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