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### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Vandermonde determinants (introduced by Alexandre-Théophile Vandermonde). Consider distinct real numbers ${{\mathbit{a}}}_{{\mathbf{0}}}{\mathbf{,}}{{\mathbit{a}}}_{{\mathbf{1}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{{\mathbit{a}}}_{\mathbf{n}\mathbf{.}}$ . We define $\left(n+1\right){\mathbf{×}}\left(n+1\right)$ the matrix${\mathbit{A}}{\mathbf{=}}\left[\begin{array}{ccc}1& 1& ....1\\ {a}_{0}& {a}_{1}& ....{a}_{n}\\ {a}_{0}^{2}& {a}_{1}^{2}& ....{a}_{1}^{2}\\ {a}_{0}^{n}& {a}_{1}^{n}& ....{a}_{n}^{n}\end{array}\right]$Vandermonde showed that ${\mathbit{d}}{\mathbit{e}}{\mathbit{t}}\mathbf{\left(}\mathbf{A}\mathbf{\right)}{\mathbf{=}}\mathbf{\prod }_{\mathbf{i}\mathbf{>}\mathbf{j}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}}\mathbf{-}{\mathbf{a}}_{\mathbf{j}}\mathbf{\right)}$ the product of all differences $\left({a}_{i}-{a}_{j}\right)$ , where exceeds j . a. Verify this formula in the case of ${\mathbit{n}}{\mathbf{=}}{\mathbf{1}}$ . b. Suppose the Vandermonde formula holds for ${\mathbit{n}}{\mathbf{=}}{\mathbf{1}}$ . You are asked to demonstrate it for n. Consider the function${\mathbf{f}}\left(t\right){\mathbf{=}}{\mathbf{det}}\left[\begin{array}{ccccc}1& 1& ...& 1& 1\\ {a}_{0}& {a}_{1}& ...& {a}_{n-1}& t\\ {a}_{0}^{2}& {a}_{1}^{2}& ...& {a}_{n-1}& {t}^{2}\\ ⋮& ⋮& ...& ⋮& ⋮\\ {a}_{0}^{n}& {a}_{1}^{n}& ...& {a}_{n-1}^{n}& {t}^{n}\end{array}\right]$Explain why f(t) is a polynomial of ${{\mathbit{n}}}^{\mathbf{t}\mathbf{h}}$degree. Find the coefficient k of ${{\mathbit{t}}}^{{\mathbf{n}}}$ using Vandermonde's formula for ${{\mathbit{a}}}_{{\mathbf{0}}}{\mathbf{,}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{,}}{{\mathbit{a}}}_{\mathbf{n}\mathbf{-}\mathbf{1}}$. Explain why role="math" localid="1659522435181" ${\mathbit{f}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{0}}\mathbf{\right)}{\mathbf{=}}{\mathbit{f}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{=}}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}{\mathbf{=}}{\mathbit{f}}\mathbf{\left(}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{\right)}{\mathbf{=}}{\mathbf{0}}$Conclude that ${\mathbit{f}}\mathbf{\left(}\mathbit{t}\mathbf{\right)}{\mathbf{=}}{\mathbit{k}}\mathbf{\left(}\mathbit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{0}}\mathbf{\right)}\mathbf{\left(}\mathbit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{1}}\mathbf{\right)}{\mathbf{.}}{\mathbf{.}}{\mathbf{.}}\mathbf{\left(}\mathbit{t}\mathbf{-}{\mathbf{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}\mathbf{\right)}$ for the scalar k you found above. Substitute ${\mathbf{t}}{\mathbf{=}}{{\mathbf{a}}}_{{\mathbf{n}}}$ to demonstrate Vandermonde's formula.

Therefore, the being the Vandermonde's determinant for , we have exactly .

$f\left(t\right)=k\prod _{i=0}^{n-1}\left(t-{a}_{i}\right)$

See the step by step solution

## Step 1: (a) By using Vandermonde’s Formula.

For $n=1$, we have a $2×2$ matrix

$A=\left[\begin{array}{cc}1& 1\\ {a}_{0}& {a}_{1}\end{array}\right]$

Using Vandermonde's formula, we have

$\prod _{i>j}\left({a}_{i}-{a}_{j}\right)={a}_{1}-{a}_{0}=detA$

## Step 2: (b) To  Find the coefficient k of tn using Vandermonde's formula.

By the Laplace expansion along the $\left(n+1\right)$ -th column, we see that f is a polynomial of n -th degree, the coefficient of being in fact the Vandermonde's determinant for $n-1$ , which is $\prod _{i,j-1i>j}^{n-1}\left({a}_{i}-{a}_{j}\right)\phantom{\rule{0ex}{0ex}}$.

For $t={a}_{m},m=0,1,...,n-1$ , the $\left(m+1\right)$ -th and the $\left(n+1\right)$ -th column will be the same, thus the determinant will be 0 . So,

$f\left({a}_{m}\right)=0,\forall m=0,1,...,n$

For k being the Vandermonde's determinant for , we have exactly $f\left(t\right)=k\prod _{i=0}^{n-1}\left(t-{a}_{i}\right)$ .