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The Vandermonde matrix is the matrix of with respect to the canonical bases of and +. Changing the basis of amounts to multiplying the Vandermonde matrix by a change-of-basis matrix M (from the right).
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
This matrix is a Vandermonde matrix over . In other words, the Reed–Solomon code is a linear code , and in the classical encoding procedure, its generator matrix is A {\displaystyle A} . Systematic encoding procedure: The message as an initial sequence of values
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.
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
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.
In algebra, the Vandermonde polynomial of an ordered set of n variables , …,, named after Alexandre-Théophile Vandermonde, is the polynomial: = < (). (Some sources use the opposite order (), which changes the sign () times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.)
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 ...