## sub-matrix determinants of a matrix and its similarity transformations

Consider $A$ to be an $n \times n$ matrix and $U$ an unitary matrix such that $UAU^{*}=T$ is triangular.

We know that $\mbox{det}(A)=\mbox{det}(UAU^{*}).$

Now consider $S$ to be $q\leq n$ arbitrary indices from the set $[n[={1,2,\ldots,n}$ and define $A_{S,\ S}$ to be the $q \times q$ square matrix removing all the rows and columns of $A$ not in the set $S$. The question is what the relation between $\mbox{det}(A_{S,\ S})$ and $\mbox{det}(UAU^{*}_{S,\ S})$ would be.

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