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An m × n rectangular Vandermonde matrix such that m ≤ n has rank m if and only if all x i are distinct. An m × n rectangular Vandermonde matrix such that m ≥ n has rank n if and only if there are n of the x i that are distinct. A square Vandermonde matrix is invertible if and only if the x i are distinct. An explicit formula for the ...
The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie. [1] There is a q-analog to this theorem called the q-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity
A matrix with the only nonzero entries on the main diagonal and the diagonals just above and below the main one. X–Y–Z matrix A generalization to three dimensions of the concept of two-dimensional array: Vandermonde matrix: A row consists of 1, a, a 2, a 3, etc., and each row uses a different variable. Walsh matrix
The matrix X on the left is a Vandermonde matrix, whose determinant is known to be () = < (), which is non-zero since the nodes are all distinct. This ensures that the matrix is invertible and the equation has the unique solution A = X − 1 ⋅ Y {\displaystyle A=X^{-1}\cdot Y} ; that is, p ( x ) {\displaystyle p(x)} exists and is unique.
This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum ( 1 / N {\displaystyle 1/{\sqrt {N}}} ) and the sign of the exponent in ω are merely conventions, and differ in some treatments.
Therefore, it is preferred in proofs and theoretical arguments. Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the Vandermonde determinant. But, as can be seen from the construction, each time a node x k changes, all Lagrange basis polynomials have to be recalculated.
Rather, the Jordan canonical form of () contains one Jordan block for each distinct root; if the multiplicity of the root is m, then the block is an m × m matrix with on the diagonal and 1 in the entries just above the diagonal. in this case, V becomes a confluent Vandermonde matrix. [2]
The Hermite interpolation problem is a problem of linear algebra that has the coefficients of the interpolation polynomial as unknown variables and a confluent Vandermonde matrix as its matrix. [3] The general methods of linear algebra, and specific methods for confluent Vandermonde matrices are often used for computing the interpolation ...