Exact calculation of some induced matrix norms
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This paper will investigate the induced matrix norms of linear operators between finite dimensional vector spaces. In the past, the closed form formulas for $\|A\|_{pq}$ were known for some $p, q = 1, 2, \infty$, but not for $\|A\|_{2\infty}$ and $\|A\|_{\infty 2}$. The formulas for these two norms and for $\|A\|_{21}$ have been derived in this paper. The computation of $\|A\|_{21}$, $\|A\|_{\infty 1}$ and $\|A\|_{\infty 2}$ is generally difficult, since they concern all the arrangements of objects. The algorithms developed here can provide the exact numerical value with the least amount of calculations.
Abstract
This paper will investigate the induced matrix norms of linear operators between finite dimensional vector spaces. In the past, the closed form formulas for $\|A\|_{pq}$ were known for some $p, q = 1, 2, \infty$, but not for $\|A\|_{2\infty}$ and $\|A\|_{\infty 2}$. The formulas for these two norms and for $\|A\|_{21}$ have been derived in this paper. The computation of $\|A\|_{21}$, $\|A\|_{\infty 1}$ and $\|A\|_{\infty 2}$ is generally difficult, since they concern all the arrangements of objects. The algorithms developed here can provide the exact numerical value with the least amount of calculations.