# Geometric interpretation of matrix minors

i was recently interested in geometric interpretation of various notions showing up in linear algebra because in most cases linear algebra with geometry 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.

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 mathoverflow.net 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:

what is the geometrical interpretation of matrix minor?

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! ;)

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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.

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i just forgot about exterior algebra… :p i just don't get it thoroughly but as far as i get it: the question i posed in your language would sound like 'how does $T$ it act on those parallelotopes…?' could you dwell little bit more on this? i must confess i'd be more satisfied seeing more school geometry here—care to translate from exterior algebra? :p – joel Sep 12 '11 at 21:06
@joel: a wedge product $v_1 \wedge ... \wedge v_k$ describes an oriented $k$-paralleletope with vertices $0, v_i, v_i + v_j, ...$ and $T$ acts on it by sending it to the paralleletope described by $T(v_1) \wedge ... \wedge T(v_k)$. Writing this in terms of sums of exterior products of a fixed basis of $W$ corresponds to slicing up the paralleletope along various axes. – Qiaochu Yuan Sep 12 '11 at 21:16
can i say then that minor of 'codegree' 1 describes measure of parallelotope's face of codimension 1 spaned by appropriate vectors? if so, what can i say about other minors in similar manner—what are principal minors then? (i'm afraid that hodge duality might spring out… ;p) tell me how to interpret laplace formula where minors show up (i won't have problems with cofactors—signs seem to come from keeping track of orientation…) probably your answer is definite but i'm still looking here for purely geometrical description which i could understand—care to give completely new answer? :p – joel Sep 13 '11 at 0:19
@joel: I don't appreciate receiving a command like "tell me X." I think this is a purely geometric description, if one just starts with a suitably geometric intuition for the exterior algebra, so perhaps you should be asking about that instead. – Qiaochu Yuan Sep 13 '11 at 1:54
i think you've helped me quite much, so please excuse my overfamiliar and presumptuous behaviour—i hadn't in mind giving orders to anybody: i just don't feel the courtesy and politeness levels of the language good enough yet.<br>mind to explore the topic little further as you proposed? :) – joel Sep 13 '11 at 9:21

One graph-theoretic interpretation of minors is given by Linstrom's lemma. http://geomblog.blogspot.com/2004/06/lindstroms-lemma.html

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"Lindström", to be precise. – Hans Lundmark Sep 13 '11 at 9:29
:-) I've always wondered how to put those and similar markings in my posts. (And how to remember the answer when I haven't used it for over a year.) – Pace Nielsen Sep 13 '11 at 15:47