proof of determinant of the Vandermonde matrix
To begin, note that the determinant of the Vandermonde matrix (which we shall denote as ‘’) is a homogeneous polynomial of order because every term in the determinant is, up to sign, the product of a zeroth power of some variable times the first power of some other variable , , the -st power of some variable and .
Next, note that if with , then because two columns of the matrix would be equal. Since is a polynomial, this implies that is a factor of . Hence,
where C is some polynomial. However, since both and the product on the right hand side have the same degree, must have degree zero, i.e. must be a constant. So all that remains is the determine the value of this constant.
One way to determine this constant is to look at the coefficient of the leading diagonal, . Since it equals 1 in both the determinant and the product, we conclude that , hence
|Title||proof of determinant of the Vandermonde matrix|
|Date of creation||2013-03-22 15:44:50|
|Last modified on||2013-03-22 15:44:50|
|Last modified by||rspuzio (6075)|