<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>alephone</author><text>&amp;quot;square matrices have as many eigenvectors as they have linearly independent dimensions; i.e. a 2 x 2 matrix would have two eigenvectors, a 3 x 3 matrix three, and an n x n matrix would have n eigenvectors, each one representing its line of action in one dimension.&amp;quot;&lt;p&gt;This is not quite right. First of all, if v is an eigenvector of A, then a*v for any non-zero a is also an eigenvector of A, so it makes no sense to say &amp;quot;there are n eigenvectors&amp;quot;. Perhaps you meant to say n linearly independent eigenvectors?&lt;p&gt;But this is also not correct. An n x n matrix does not necessarily have to have n linearly independent eigenvectors; consider for instance A = [1, 1; 0, 1], which has dim N(A - I) = 1 (hence 1 linearly independent eigenvector). Basically, not every matrix is diagonalizable. This example also shows that rank(A) is also not necessarily equal to the number of linearly independent eigenvectors (is that what you meant by linearly independent dimensions?).&lt;p&gt;A more correct statement would be to say every symmetric matrix has n linearly independent eigenvectors (and its eigenvectors can be chosen to form an orthonormal basis for R^n).</text><parent_chain></parent_chain></comment><story><title>A Beginner’s Guide to Eigenvectors, Covariance, PCA and Entropy</title><url>http://deeplearning4j.org/eigenvector</url></story>
<instructions>Your goal is to analyze the following comment and estimate how highly it will be upvoted by the Hacker News community.</instructions><comment><author>tsumnia</author><text>You linked it on the bottom of the page, but how do you think the article compares with &lt;a href=&quot;http:&amp;#x2F;&amp;#x2F;www.cs.otago.ac.nz&amp;#x2F;cosc453&amp;#x2F;student_tutorials&amp;#x2F;principal_components.pdf&quot; rel=&quot;nofollow&quot;&gt;http:&amp;#x2F;&amp;#x2F;www.cs.otago.ac.nz&amp;#x2F;cosc453&amp;#x2F;student_tutorials&amp;#x2F;principa...&lt;/a&gt; ?&lt;p&gt;I&amp;#x27;m planning to read through it a little more this weekend, but I remember using the PDF and a whiteboard back in the day to understand PCA.</text><parent_chain></parent_chain></comment><story><title>A Beginner’s Guide to Eigenvectors, Covariance, PCA and Entropy</title><url>http://deeplearning4j.org/eigenvector</url></story>