MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How to find how many zero principal minors does a matrix have? Is there any easy way to compute principal minors?

share|cite|improve this question
Please read MathOverflow is for questions about research-level mathematics. Perhaps would be a better venue for this question. – David Roberts Jun 20 '11 at 4:21

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.

share|cite|improve this answer
Nice answer, but you might want to define "unimodular"... – Igor Rivin Jun 20 '11 at 13:44

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.