Timeline for Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?
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| Feb 23, 2022 at 17:43 | comment | added | Richard Stanley | In fact, the eigenvalues of $C_p(A)$ consist of products of $p$ of the eigenvalues of $A$. Let $M$ be the $n\times {n\choose p}$ incidence matrix between elements of $[n]=\{1,\dots,n\}$ and $p$-element subsets of $[n]$. The Smith normal form of $M$ has diagonal entries $p$ (once) and 1 ($n-1$ times), a special case of Theorem 3.6 of people.clas.ufl.edu/sin/files/snf.pdf. This implies that the $p$th powers of the eigenvalues of $A$ can be written as Laurent monomials in the eigenvalues of $C_p(A)$. | |
| Feb 21, 2022 at 19:50 | history | edited | LSpice | CC BY-SA 4.0 |
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| Feb 21, 2022 at 17:09 | history | answered | Nathaniel Johnston | CC BY-SA 4.0 |