Properties
Using the Leibniz formula for the determinant,
where Sn denotes the set of permutations of, and sgn(σ) denotes the signature of the permutation σ, we can rewrite the Vandermonde determinant as
The Vandermonde polynomial (multiplied with the symmetric polynomials) generates all the alternating polynomials.
If m ≤ n, then the matrix V has maximum rank (m) if and only if all αi are distinct. A square Vandermonde matrix is thus invertible if and only if the αi are distinct; an explicit formula for the inverse is known.
Read more about this topic: Vandermonde Matrix
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)