https://www.chenk.top/en/tags/spectral-radius/Recent content in Spectral Radius on Chen Kai BlogHugoenWed, 05 Mar 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/10-matrix-norms-and-condition-numbers/Wed, 05 Mar 2025 09:00:00 +0000https://www.chenk.top/en/linear-algebra/10-matrix-norms-and-condition-numbers/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/linear-algebra/10-matrix-norms-and-condition-numbers/illustration_1.png" alt="Essence of Linear Algebra (10): Matrix Norms and Condition Numbers — Is Your Linear System Healthy? — Chapter overview" loading="lazy" decoding="async" class="content-image"> </figure> </p> <hr> <h2 id="the-question-that-haunts-engineers" class="heading-anchor">The Question That Haunts Engineers<a href="#the-question-that-haunts-engineers" class="heading-link" aria-label="Permalink to this section" title="Copy link to this section">#</a> </h2><p>The equations are right. The algorithm is right. So why is the computed answer completely wrong?</p> <p>The culprit is usually a single number called the <strong>condition number</strong>. It measures how <em>sensitive</em> a linear system is — whether a tiny wobble in the input gets amplified into a catastrophic error in the output. To talk about condition numbers we first need a way to measure the &ldquo;size&rdquo; of vectors and matrices. That is what norms do.</p>