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

These matrices are useful in polynomial interpolation, since solving the system of linear equations Vu = y for u with V an n × n Vandermonde matrix is equivalent to finding the coefficients u_{j} of the polynomial
_{i} at α_{i}.
The Vandermonde determinant plays a central role in the Frobenius formula, which gives the character of conjugacy classes of representations of the symmetric group. ## See also

## References

- $V=begin\{bmatrix\}$

- $V\_\{i,j\}\; =\; alpha\_i^\{j-1\}$

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

- $det(V)\; =\; prod\_\{1le\; i\}\; (alpha\_j-alpha\_i).\; math>$

The above determinant is sometimes called the discriminant, although many sources, including this article, refer to the discriminant as the square of this determinant.

Using the Leibniz formula

- $det(V)\; =\; sum\_\{sigma\; in\; S\_n\}\; sgn(sigma)\; ,\; alpha\_1^\{sigma(1)-1\}\; cdots\; alpha\_n^\{sigma(n)-1\},$

- $prod\_\{1le\; i\}\; (alpha\_j-alpha\_i)="sum\_\{sigma"\; in\; s\_n\}\; sgn(sigma)\; ,\; alpha\_1^\{sigma(1)-1\}\; cdots\; alpha\_n^\{sigma(n)-1\},\; math>$

where S_{n} denotes the set of permutations of {1, 2, ..., n}, and sgn(σ) denotes the signature of the permutation σ.

If m≤n, then the matrix V has maximum rank (m) if and only if all α_{i} are distinct.

When two or more α_{i} are equal, the corresponding polynomial interpolation problem (see below) is underdetermined. In that case one may use a generalization called confluent Vandermonde matrices, which makes the matrix non-singular while retaining most properties. If α_{i} = α_{i + 1} = ... = α_{i+k} and α_{i} ≠ α_{i − 1}, then the (i + k)th row is given by

- $V\_\{i+k,j\}\; =\; begin\{cases\}\; 0,\; \&\; mbox\{if\; \}\; j\; le\; k;\; frac\{(j-1)!\}\{(j-k-1)!\}\; alpha\_i^\{j-k-1\},\; \&\; mbox\{if\; \}\; j\; >\; k.\; end\{cases\}$

- $P(x)=sum\_\{j=0\}^\{n-1\}\; u\_j\; x^j$

When the values $alpha\_k$ range over powers of a finite field, then the determinant has a number of interesting properties: for example, in proving the properties of a BCH code.

Confluent Vandermonde matrices are used in Hermite interpolation.

A commonly known special Vandermonde matrix is the discrete Fourier transform matrix, where the numbers α_{i} are chosen equal to the m different mth roots of unity.

The Vandermonde matrix diagonalizes a companion matrix.

- Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, (1991) Cambridge University Press. See Section 6.1.
- Lecture 4 reviews the representation theory of symmetric groups, including the role of the Vandermonde determinant.

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Last updated on Thursday September 18, 2008 at 05:58:18 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday September 18, 2008 at 05:58:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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