Math 2.0

Linear and multilinear algebra; matrix theory Determinants Theorem

Vandermonde determinant

Builds on Determinant

Statement

Let KK be a commutative ring, let n≥1n \ge 1, and let x1,…,xn∈Kx_1, \dots, x_n \in K. Let VV be the n×nn \times n Vandermonde matrix with entries Vij=xi j−1V_{ij} = x_i^{\,j-1}. Then det⁡V=∏1≤i<j≤n(xj−xi),\det V = \prod_{1 \le i < j \le n} (x_j - x_i), where det⁡\det is the determinant.

Remarks

Over a field, det⁡V≠0\det V \ne 0 if and only if the xix_i are pairwise distinct. Then, for all y1,…,yny_1, \dots, y_n, there is exactly one polynomial pp of degree less than nn with p(xi)=yip(x_i) = y_i for all ii. For n=2n = 2, the matrix is (1x11x2)\begin{pmatrix} 1 & x_1 \\ 1 & x_2 \end{pmatrix} with determinant x2−x1x_2 - x_1.

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  1. Identity matrix object
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  9. Vandermonde determinant statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Vandermonde_matrix, Determinant
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Tag0P84B554AS
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Kindstatement, theorem
WritingW0 stub
MSC15A15, 15B05
StatusThis copy of the site has no build data. Run m2 status 0P84B554AS.
UsesDeterminant

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