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Let $L$ be an $n\times n$ matrix and $W= \operatorname{diag}(w_1,\cdots,w_n)$. Show that $\det(I-WL)= \sum_{S\subset[n]} \det(L[S])(-1)^{|S|}w^S$. Where $I$ is the $n\times n$ identity matrix, $L[S]$ is the principal submatrix of $L$ whose rows and columns are indexed by $S$ and $w^S = \prod_{i\in S} w_i$.

I am thinking there is some way to apply Cauchy-Binet in this case.

Bernard
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Sudipta Roy
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  • First prove that $\det\left(I + A\right) = \sum\limits_{S \subseteq \left[n\right]} \det\left(A\left[S\right]\right)$ for any $n \times n$-matrix $A$. (This is Proposition 1 in https://math.stackexchange.com/a/1752326/ where I give a reference to a detailed proof.) Now apply it to $A = - WL$ and simplify $\det\left(\left(WL\right)\left[S\right]\right)$ to $\left(-1\right)^{\left|S\right|} w^S \det\left(L\left[S\right]\right)$ (because $- WL$ is obtained from $L$ by multiplying each row with one of the $-w_i$). – darij grinberg Aug 04 '18 at 16:52
  • @darij-grinberg What about making your comment an answer? – J.-E. Pin Aug 30 '18 at 02:55
  • @J.-E.Pin whenever I find the time (hopefully soon). – darij grinberg Aug 31 '18 at 21:39

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