⏚ Antoine Chambert-Loir<p>Evaluating the operator norms of matrices<br>New blog post on Freedom Math Dance</p><p>Let E and F be normed vector spaces, over the real or complex numbers, and let u :E→F be a linear map. The continuity of uu is proved to be equivalent to the existence of a real number cc such that ∥u(x)∥≤c∥x∥ for every x∈E, and the least such real number is called the operator norm of uu; we denote it by ∥u∥. It defines a norm on the linear space L(E;F) of continuous linear maps from E to F and as such is quite important. When E=F, it is also related to the spectrum of uu and is implicitly at the heart of criteria for the Gershgorin criterion for localization of eigenvalues.</p><p>However, even in the simplest cases of matrices, its explicit computation is not trivial at all, and as we'll see even less trivial than what is told in algebra classes, as I learned by browsing Wikipedia when I wanted to prepare a class on the topic.</p><p>Follow the link to read… → <br><a href="https://freedommathdance.blogspot.com/2024/04/evaluating-operator-norms-of-matrices.html" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">freedommathdance.blogspot.com/</span><span class="invisible">2024/04/evaluating-operator-norms-of-matrices.html</span></a></p><p><a href="https://mathstodon.xyz/tags/LinearAlgebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>LinearAlgebra</span></a> <a href="https://mathstodon.xyz/tags/ComplexityTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>ComplexityTheory</span></a></p>