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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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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