Geometric interpretation of matrix minors - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:01:53Z http://mathoverflow.net/feeds/question/75241 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors Geometric interpretation of matrix minors joel 2011-09-12T18:01:32Z 2011-09-12T19:04:59Z <p>i was recently interested in geometric interpretation of various notions showing up in linear algebra because in most cases <em>linear algebra with geometry</em> courses focus too much on linear algebra not giving rationale or intuitions behind ideas hinting that lecturer doesn't have much idea about mathematics involved staying only on the surface of whole matter (maybe only due to didactic purposes). it's quite funny that answer doesn't show up anywhere on the network or in textbooks: in most cases i find this style very poor showing somewhat that author of the book / script might not have full grasp of it neither.</p> <p>this can only mean that it hasn't got much of geometric interpretation which i'm eager to reject because i haven't had found in my first handbooks any intuitions regarding determinant or trace (those problems where addressed on <code>mathoverflow.net</code> already: volume of oriented parallelotope and measure of change of edges in respect to edges of unit hypercube, respectively) but what i'm interested most at the moment is:</p> <blockquote> <p>what is the geometrical interpretation of matrix minor?</p> </blockquote> <p>i can only guess that it gives measure of some object of lower dimension for parallelotope, it sides for minor of highest but one degree; i could only guess that minors are projections on respective axes or something but i'm not sure of my interpretation. probably laplace expansion could give decisive answer but i also fail to interpret it properly; other clue might lie in expanded characteristic polynomial as for then it's coefficients are sums of all principial minors (of degree equal to degree of the term as far as i know). one can also find them in search of invertible coordinates in inverse and implicit function theorems. you're welcome to give interpretation of those using given intuitions! ;)</p> http://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors/75246#75246 Answer by Pace Nielsen for Geometric interpretation of matrix minors Pace Nielsen 2011-09-12T18:53:24Z 2011-09-12T18:53:24Z <p>One graph-theoretic interpretation of minors is given by Linstrom's lemma. <a href="http://geomblog.blogspot.com/2004/06/lindstroms-lemma.html" rel="nofollow">http://geomblog.blogspot.com/2004/06/lindstroms-lemma.html</a></p> http://mathoverflow.net/questions/75241/geometric-interpretation-of-matrix-minors/75248#75248 Answer by Qiaochu Yuan for Geometric interpretation of matrix minors Qiaochu Yuan 2011-09-12T19:04:59Z 2011-09-12T19:04:59Z <p>A minor is a matrix element of the action of a linear transformation $T : V \to W$ on exterior powers $\Lambda^k(T) : \Lambda^k(V) \to \Lambda^k(W)$. The geometric entity is the entire linear transformation $\Lambda^k(T)$ (or its trace, which is what appears in the characteristic polynomial), which describes how $T$ acts on oriented paralleletopes of various intermediate dimensions. </p>