How to find how many zero principal minors does a matrix have? Is there any easy way to compute principal minors?
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1$\begingroup$ Please read mathoverflow.net/howtoask. MathOverflow is for questions about research-level mathematics. Perhaps math.stackexchange.com would be a better venue for this question. $\endgroup$– David Roberts ♦Commented Jun 20, 2011 at 4:21
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For the incidence matrix $\partial(G)$ of a graph $G$ the answer is easy, as this amounts to knowing how many spanning trees $G$ possess. One can use the matrix-tree theorem to compute the answer by evaluating the determinant of the (reduced) Laplacian matrix $(\partial(G)\partial(G)^T)^{vv}$ of $G$.
For a full rank unimodular matrix $M$ (i.e., all maximal minors are $\pm 1$ or zero) with rational entries there is a related "matrix-tree" type theorem: The number of maximal non-zero minors of $M$ is $\det(M M^T)$. This is probably most useful when the matrix handed to you is a priori unimodular, as above.
For a matrix $M$ which fails to be unimodular the number $\det(MM^T)$ gives an upper bound.
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$\begingroup$ Nice answer, but you might want to define "unimodular"... $\endgroup$ Commented Jun 20, 2011 at 13:44