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In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with a geometric progression in each row, i.e; Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...
Alexandre-Théophile Vandermonde (28 February 1735 -1 January 1796) was a French musician and chemist who worked with Bezout and Lavoisier; his name is now principally associated with determinant theory in mathematics. ...
For the square matrix section, see square matrix. ...
In mathematics, a geometric progression (also inaccurately known as a geometric series, see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...
or for all indices i and j. (Some authors use the transpose of the above matrix.) See transposition for meanings of this term in telecommunication and music. ...
The determinant of an n × n Vandermonde matrix can be expressed as: In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
The above determinant is sometimes called the discriminant, although many authors, including Wikipedia, refer to the discriminant as the square of this determinant. In mathematics, a polynomial P(T) has a discriminant, which is a polynomial function of its coefficients, and discriminates the case of a multiple root (for which the graph of P(x) would touch the x-axis). ...
Using the Leibniz formula for the determinant, we can rewrite this formula as In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
where Sn denotes the set of permutations of {1, 2, ..., n}, and sgn(σ) denotes the signature of the permutation σ. In mathematics, especially in abstract algebra and related areas, a permutation is a bijection, from a finite set X onto itself. ...
In mathematics, the permutations of a finite set (i. ...
If m≤n, then the matrix V has maximum rank (m) if and only if all αi are distinct. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ...
When two or more αi are equal, the corresponding polynomial interpolation problem is ill-posed. In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix positive definite while retaining most properties. If αi = αi+1 = ... = αi+k and αi ≠ αi-1, then the (i + k)th row is given by In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
The above formula for confluent Vandermonde matrices can be readily derived by letting two parameters αi and αj go arbitrarily close to each other. The difference vector between the rows corresponding to αi and αj scaled to a constant yields the above equation (for k=1). Similarly, the cases k>1 are obtained by higher order differences. Consequently, the confluent rows are derivatives of the original Vandermonde row.
Applications These matrices are useful in polynomial interpolation, since solving the system of linear equations Vu=y for u with V the n × n Vandermonde matrix is equivalent to finding the coefficients uj of the polynomial In the mathematical subfield of numerical analysis, polynomial interpolation is the interpolation of a given data set by a polynomial. ...
In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...
In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ...
of degree ≤ n−1 which has the values yi at αi.
The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group. In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...
In mathematics, the permutation group SN has order N!. Its conjugacy classes are labeled by partitions of N. Therefore according to the representation theory of a finite group the number of inequivalent irreducible representations is equal to the number of partitions of N. They are labeled by Young diagrams. ...
In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...
When the values αk range over powers of a finite field, then the determinant is more commonly known as the Moore determinant, which has a number of interesting properties. In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...
Confluent Vandermonde matrices are used in Hermite interpolation. Hermite interpolation is a method closely related to the Newton divided difference method of interpolation in numerical analysis, that allows us to consider given derivatives at data points, as well as the data points themselves. ...
A commonly known special Vandermonde matrix is the discrete Fourier transform matrix. In mathematics, the discrete Fourier transform (DFT), sometimes called the finite Fourier transform, is a Fourier transform widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, solve partial differential equations, and to perform other operations such as convolutions. ...
See also In numerical analysis, a Lagrange polynomial, named after Joseph Louis Lagrange, is the interpolation polynomial for a given set of data points in the Lagrange form. ...
In mathematics, the Wronskian is a function named after Polish mathematician Josef Hoene-Wronski, especially important in the study of differential equations. ...
References - Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, (1991) Cambridge University Press. See Section 6.1.
- William Fulton and Joe Harris, Representation Theory, A First Course (1991) Springer Verlag New York, ISBN 0-387-974495-4 Chapter 4 reviews the representation theory of symmetric groups, including the role of the Vandermonde determinant.
- David Goss, Basic Structures of Function Field Arithmetic (1996) Springer Verlag New York, ISBN 3-540-63541-6 Chapter 1 reviews the Moore determinant
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