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# Vandermonde行列式

A=\left(\begin{array}{cccc}1&1&\cdots&1\\x_1&x_2&\cdots&x_n\\\vdots&\vdots&\ddots&\vdots\\x_1^{n-1}&x_2^{n-1}&\cdots&x_n^{n-1}\end{array}\right)


Vandermonde行列の行列式は各行の公比の差積に等しく，以下のように表されます．

det(V)=\prod_{1\leq{i}<j\leq{n}}(x_j-x_i)=(-1)^{n(n-1)/2}\prod_{1\leq{i}<j\leq{n}}(x_i-x_j)


（証明）

(i) $n=2$のとき

A=\left|\begin{array}{cc}1&1\\x_1&x_2\end{array}\right|=x_2-x_1


より成り立つ．

(ii) $n=k-1$のときに成り立つと仮定する．$n=k$のときを考えると，その行列式は

det(A)=\left|\begin{array}{cccc}1&1&\cdots&1\\x_1&x_2&\cdots&x_k\\x_1^2&x_2^2&\cdots&x_k^2\\\vdots&\vdots&\ddots&\vdots\\x_1^{k-1}&x_2^{k-1}&\cdots&x_k^{k-1}\end{array}\right|


これから，$2\sim{k-1}$行までの各行を$x_1$倍して次の行から引くと，

det(A)=\left|\begin{array}{cccc}1&1&\cdots&1\\0&x_2-x_1&\cdots&x_k-x_1\\0&x_2(x_2-x_1)&\cdots&x_k(x_k-x_1)\\\vdots&\vdots&\ddots&\vdots\\0&x_2^{k-2}(x_2-x_1)&\cdots&x_k^{k-2}(x_k-x_1)\end{array}\right|\\=\left|\begin{array}{ccc}x_2-x_1&\cdots&x_k-x_1\\x_2(x_2-x_1)&\cdots&x_k(x_k-x_1)\\\vdots&\ddots&\vdots\\x_2^{k-2}(x_2-x_1)&\cdots&x_k^{k-2}(x_k-x_1)\end{array}\right|\\=(x_2-x_1)\cdots(x_k-x_1)\left|\begin{array}{ccc}1&\cdots&1\\x_2&\cdots&x_k\\\vdots&\ddots&\vdots\\x_2^{k-2}&\cdots&x_k^{k-2}\end{array}\right|\\=(x_2-x_1)\cdots(x_k-x_1)\prod_{1\leq{i}<j\leq{k-1}}(x_j-x_i)\\=\prod_{1\leq{i}<j\leq{k}}(x_j-x_i)


det(A)=\left|\begin{array}{cccc}1&1&1&1\\2&3&5&4\\4&9&25&16\\8&27&125&64\end{array}\right|


として，Vandermonde行列式の式およびJuliaの関数による行列式の出力を比較します．

# Juliaで実験


using LinearAlgebra

A = [1 1 1 1;
2 3 5 4;
4 9 25 16;
8 27 125 64]

function vandermonde(A)
det = 1
for j in 2:4
for i in 1:j-1
det *= (A[2,j]-A[2,i])
end
end

det
end

println("vandermonde: ", vandermonde(A))
println("det(A): ", det(A))


vandermonde: -12
det(A): -11.999999999999993


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