Vandermonde matrix

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In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row, i.e., an m × n matrix

or

for all indices i and j.[1] (Some authors use the transpose of the above matrix.)

The determinant of a square Vandermonde matrix (where m = n) can be expressed as

This is called the Vandermonde determinant or Vandermonde polynomial. If all the numbers are distinct, then it is non-zero.

The Vandermonde determinant was sometimes called the discriminant, although, presently, the discriminant is the square of the Vandermonde determinant. The Vandermonde determinant is an alternating form , meaning that exchanging two changes the sign, while permuting the by an even permutation does not change the value of the determinant. It thus depends on the choice of an order on the , while its square, the discriminant, does not depend on any order, and this implies, by Galois theory, that the discriminant is a polynomial function of the coefficients of the polynomial that has the as roots.

Proof[edit]

The main property of a square Vandermonde matrix

is that its determinant has the simple form

This may be proved either by using properties of polynomials or elementary row and column operations. The former is simpler but it is non-constructive and uses unique factorization property of multivariate polynomials. The latter is constructive and more elementary, at the price of being more complicated. A third proof, based on Gaussian elimination, is sketched. It is still more complicated, if written in details, but provides the U-part of the LU decomposition of Vandermonde matrices.

Using polynomial properties[edit]

By Leibniz formula, det(V) is a polynomial in the with integer coefficients. All entries of the ith column have the total degree i – 1. Thus, again by Leibniz formula, all terms of the determinant have the total degree

(that is the determinant is a homogeneous polynomial of this degree).

If, for ij, one substitutes for , one gets a matrix with two equal columns, which has thus a zero determinant. Thus, by the factor theorem, is a divisor of det(V). By unique factorization property of multivariate polynomials, the product of all divides det(V), that is

where Q is a polynomial. As the product of all and det(V) have the same degree the polynomial Q is, in fact, a constant. This constant is one, because the product of the diagonal entries of V is which is also the monomial that is obtained by taking the first term of all factors in This proves that

By row and columns operations[edit]

This second proof is based on the fact that, if one adds to a row (or a column) of a matrix the product by a scalar of another row (or column), the determinant remains unchanged.

If one subtracts the first row of V to all the other rows, the determinant is not changed, and the new matrix has the form

where is a row matrix, is a column of zeros, and A is a square matrix, such that

The entry of the (i – 1)th row and the (j – 1)th column of A (that is the ith row and the jth column of the whole matrix) is

Dividing out from the (i – 1)th row of A, for i = 2, ..., n, one gets a matrix B such that

The coefficient of the (i – 1)th row and the (j – 1)th column of B is

for i = 2, ..., n, and setting

Thus, subtracting, for j running from n down to 2, the (j – 2)th column of B multiplied by from the (j – 1)th column, one gets a (n – 1) × (n – 1) Vandermonde matrix in which has the same determinant as B. Iterating this process on this smaller Vandermonde matrix, one gets eventually the desired expression of det(V) as the product of the

By Gaussian elimination, U-part of LU decomposition[edit]

The determinant of Vandermonde matrices may also be computed using Gaussian elimination. This provides an explicit form of the upper triangular matrix of the LU decomposition of the matrix. For this computation one uses only the elementary row operations consisting of adding to a row a scalar multiple of a preceding row. This means than one multiplies the matrix by a lower triangular matrix with a diagonal consisting only of 1. In particular, the determinant is not changed by these transformations.

Applying Gaussian elimination to a square Vandermonde matrix, one gets eventually an upper triangular matrix

which has the same determinant as V.

A proof by induction on the steps of Gaussian elimination allows showing that, for 1 ≤ ijn, one has

where is an abbreviation for , and is the sum of all monomials of degree d in In particular, the first rows of V and W are equal, and the factor equals 1 for the entries of the diagonal (since 1 is the only monomial of degree 0).

A key ingredient of this proof is that, for k > i

For the recursion, one has to describe the matrix at each step of the Gaussian elimination. Let be the entry of the ith row and jth column of this variable matrix. Before the ith step, the entries that belong either to the i first rows or the i – 1 first columns have the values that they will have at the end of Gaussian elimination, and, for ijn and ihn, one has

This is true before the first step, and one has to prove that this remains true during Gaussian elimination. The ith step does not change the i first rows nor the i – 1 first columns. It changes to zero for i < hn. For i < jn and i < hn, it changes into That is, the new is

This shows that the structure of the is kept during Gaussian elimination, and thus the form of W.

It follows from the structure of W that is the product of the diagonal entries of W, which proves again the formula for the determinant of a Vandermonde matrix.

Resulting properties[edit]

A m × n rectangular Vandermonde matrix such that mn has maximum rank m if and only if all xi are distinct.

A m × n rectangular Vandermonde matrix such that mn has maximum rank n if and only if there are n of the xi that are distinct.

A square Vandermonde matrix is invertible if and only if the xi are distinct. An explicit formula for the inverse is known.[2][3][4]

Applications[edit]

The Vandermonde matrix evaluates a polynomial at a set of points; formally, it is the matrix of the linear map that maps the vector of coefficients of a polynomial to the vector of the values of the polynomial at the values appearing in the Vandermonde matrix. The non-vanishing of the Vandermonde determinant for distinct points shows that, for distinct points, the map from coefficients to values at those points is a one-to-one correspondence, and thus that the polynomial interpolation problem is solvable with a unique solution; this result is called the unisolvence theorem.

This may be useful in polynomial interpolation, since inverting the Vandermonde matrix allows expressing the coefficients of the polynomial in terms of the and the values of the polynomial at the However, the interpolation polynomial is generally easier to compute with the Lagrange interpolation formula,[5] Which may be used for deriving a formula for the inverse of a Vandermonde matrix.

The Vandermonde determinant is used in the representation theory of the symmetric group.[6]

When the values belong to a finite field, then the Vandermonde determinant is also called Moore determinant, and has specific properties that are used, for example for the theory of BCH code and Reed–Solomon error correction codes.

The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix), where the numbers αi are chosen to be roots of unity.

Confluent Vandermonde matrices[edit]

As described before, a Vandermonde matrix describes the linear algebra interpolation problem of finding the coefficients of a polynomial of degree based on the values , where are distinct points. If are not distinct, then this problem does not have a unique solution (which is reflected by the fact that the corresponding Vandermonde matrix is singular). However, if we give the values of the derivatives at the repeated points, then the problem can have a unique solution. For example, the problem

where is a polynomial of degree , has a unique solution for all . In general, suppose that are (not necessarily distinct) numbers, and suppose for ease of notation that equal values come in continuous sequences in the list. That is

where and are distinct. Then the corresponding interpolation problem is

And the corresponding matrix for this problem is called a confluent Vandermonde matrices. In our case (which is the general case, up to permuting the rows of the matrix) the formula for it is given as follows: if , then for some (unique) (we consider ). Then, we have

This generalization of the Vandermonde matrix makes it non-singular (such that there exists a unique solution to the system of equations) while retaining most properties of the Vandermonde matrix. Its rows are derivatives (of some order) of the original Vandermonde rows.

Another way to receive this formula is to let some of the 's go arbitrarily close to each other. For example, if , then letting in the original Vandermonde matrix, the difference between the first and second rows yields the corresponding row in the confluent Vandermonde matrix. This allows us to link the generalized interpolation problem (given value and derivatives on a point) to the original case where all points are distinct: Being given is similar to being given where is very small.

See also[edit]

References[edit]

  1. ^ Roger A. Horn and Charles R. Johnson (1991), Topics in matrix analysis, Cambridge University Press. See Section 6.1.
  2. ^ Turner, L. Richard. Inverse of the Vandermonde matrix with applications (PDF).
  3. ^ Macon, N.; A. Spitzbart (February 1958). "Inverses of Vandermonde Matrices". The American Mathematical Monthly. The American Mathematical Monthly, Vol. 65, No. 2. 65 (2): 95–100. doi:10.2307/2308881. JSTOR 2308881.
  4. ^ Inverse of Vandermonde Matrix (ProofWiki)
  5. ^ Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 2.8.1. Vandermonde Matrices". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8
  6. ^ Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Lecture 4 reviews the representation theory of symmetric groups, including the role of the Vandermonde determinant.

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