VANDERMONDE'S DETERMINATS

This quiz is about Vandermonde's determinants.

Define the two matrices (written as tables for convenience):

Wn(x1, x2, ... . xn) =

 1 1 1 ... 1 x12 x22 x32 ... xn2 x13 x23 x33 ... xn3 ... ... ... ... ... x1n x2n x3n ... xnn

An(x; x1, x2, ... . xn) =
 x1 + x x2 + x x3 + x ... xn + x x12 x22 x32 ... xn2 x13 x23 x33 ... xn3 ... ... ... ... ... x1n x2n x3n ... xnn

Note that the VanderMonde matrix is Vn(x1, x2, ... . xn) = An(0; x1, x2, ... . xn).

Prove that the determinants of these two matrices are

| Wn | = | Vn |   ∑k=1 .. n (1/xn)

| An(x) | = | Vn |   { 1 + ∑k=1 .. n (x/xn) }

These results can be generalized. Define the two matrices (again written as tables for convenience):

Wnk(x1, x2, ... . xn) =

 1 1 1 ... 1 x1 x2 x3 ... xn x12 x22 x32 ... xn2 ... ... ... ... ... x1k-1 x2k-1 x3k-1 ... xnk-1 x1k+1 x2k+1 x3k+1 ... xnk+1 ... ... ... ... ... x1n x2n x3n ... xnn

Ank(x; x1, x2, ... . xn) =
 x1 + x x2 + x x3 + x ... xn + x x1 + x x2 + x x3 + x ... xn + x x12 + x x22 + x x32 + x ... xn2 + x ... ... ... ... ... x1k + x x2k + x x3k + x ... xnk + x x1k+1 x2k+1 x3k+1 ... xnk+1 ... ... ... ... ... x1n x2n x3n ... xnn

Note:
Wn1 = Wn
An1(x) = An(x)
Ank(0) = Vn .

Prove that the determinants of these two matrices are (note that   |Wn0 | = | Vn | )

| Wnk |   =   | Vn |   ∑ j1 < j2 < ... < jk (xj1   xj2   ...   xjk)-1

| Ank(x) |   =   ∑ j=0 .. k xj |Wnj|

Marco Corvi - 2003