Vandermonde matrix


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Vandermonde matrix

[′van·dər‚mȯnd ‚mā·triks]
(mathematics)
A matrix in which each entry in the first row is 1, and each entry in the i th row is the corresponding entry in the second row to the (i - 1) power.
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12, in a single snapshot condition, for the Root-MUSIC algorithm, the decrease of the number of snapshots makes the rank of Vandermonde matrix obtained from eigen-decomposition deficient.
Let X [member of] [C.sub.MxNx2] be a tensor with matrix representation [bar.Y] = (A [dot encircle] [S.sub.0])[[PHI].sup.T], where A [member of] [C.sup.MxK] is a Vandermonde matrix with distinct generators, [S.sub.0] [member of] [R.sup.NxK] and [PHI] [member of] [C.sup.2 x K].
(1) For 1 [less than or equal to] j [less than or equal to] N, the matrix [H.sub.ij] is a Vandermonde matrix, where the jth row of U in U is replaced by the ith row of V.
The Vandermonde matrix elements are calculated in the nodes [[xi].sup.j] of the element [V.sup.TET.sub.m,j] = [[psi].sup.TET.sub.m]([[xi].sup.j]), 1 [less than or equal to] m, j [less than or equal to] [N.sup.TET.sub.[phi]].
where [C.sub.p] is the scaled Vandermonde matrix [mathematical expression not reproducible] is the shifting matrix defined by [K.sub.p] = [0 [e.sub.1] ...
For example, a possible systematic solution would be the generalization of the Vandermonde matrix to any hyperoperation of rank r.
< [i.sub.t] [less than or equal to]n, Vandermonde matrix M is generated by
From (5), it is noted that Z(r), as a Vandermonde matrix, is of full column rank with L linearly independent rows.
Denote V(t) as the K x L Vandermonde matrix with klth element given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where t = {[[tau].sub.1], ..., [[tau].sub.L]} is the vector of the unknown time delays.
In fact, doing so will lead to a highly ill-posed matrix with a high-order function (the Vandermonde matrix), which has been described in [20].
note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is not the form of the generalized Vandermonde matrix.
Matrix G: Vandermonde matrix with entries [a.sub.ij] = [i.sup.j].