 | If x_1,x_2,…,x_n are numbers,then show by induction that
|1,x_1,…,x_1^(n-1)|
|1,x_2,…,x_2^(n-1)|
:
|1,x_n,…,x_n^(n-1)|
=Π[i<j](x_j-x_i)
the symbol on the right meaning that it is the product of all terms x_j-x_i with i<j and i,j integers from 1 to n.
This determinant is called the Vandermonde determinant V_n. To od the induction easily,multiply each column by x_1 and subtract it from the next column on the right,starting from the right-hand side.You will find that
V_n=(x_n-x_1)…(x_2-x_1)V_(n-1).
というヴァンデルモンドの行列式の証明問題です。
参考書に
|1,x_1,…,x_1^(n-1)|
|1,x_2,…,x_2^(n-1)|
:
|1,x_n,…,x_n^(n-1)|
=(-1)^(n(n-1)/2)Π[1≦i<j≦n](x_j-x_i)
が証明されてたのですがこれから
|1,x_1,…,x_1^(n-1)|
|1,x_2,…,x_2^(n-1)|
:
|1,x_n,…,x_n^(n-1)|
=Π[i<j](x_j-x_i)
と言う風にどうすれば(-1)^(n(n-1)/2)部分を消去できますでしょうか?
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