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linear algebra – Partial Vandermonde circulant determinant expression Answer

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linear algebra – Partial Vandermonde circulant determinant expression

The “closest” figure you may anticipate is the well-known determinant formulation for tridiagonal matrices, which in your illustration can breathe written as
$$
Delta =
det
left[
begin{pmatrix}
-x_n^{-n} & 0
0 & 1
end{pmatrix}
+
begin{pmatrix}
1 & 0
0 & -x_1^{n}
end{pmatrix}
mathbf T_n mathbf T_{n-1} cdots mathbf T_2 mathbf T_1
right] ,,prod_{ok=1}^n (-x_k^{ok+1}),,
$$

with switch matrix
$$
mathbf T_k =
commence{pmatrix}
-x_k^{-1} & -x_k^{-2}
1 & 0
aim{pmatrix}.
$$

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