From Simple English Wikipedia, the free encyclopedia
In Linear Algebra , a Vandermonde matrix , named after Alexandre-Théophile Vandermonde , is a
(
m
+
1
)
×
(
n
+
1
)
{\displaystyle (m+1)\times (n+1)}
matrix with the form:
V
=
V
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0
,
x
1
,
⋯
,
x
m
)
=
[
1
x
0
x
0
2
…
x
0
n
1
x
1
x
1
2
…
x
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n
1
x
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x
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2
…
x
2
n
⋮
⋮
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1
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…
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]
{\displaystyle V=V(x_{0},x_{1},\cdots ,x_{m})={\begin{bmatrix}1&x_{0}&x_{0}^{2}&\dots &x_{0}^{n}\\1&x_{1}&x_{1}^{2}&\dots &x_{1}^{n}\\1&x_{2}&x_{2}^{2}&\dots &x_{2}^{n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\1&x_{m}&x_{m}^{2}&\dots &x_{m}^{n}\end{bmatrix}}}
with entries
V
i
,
j
=
x
i
j
{\displaystyle V_{i,j}=x_{i}^{j}}
, the j th power of the number
x
i
{\displaystyle x_{i}}
, for all indices
i
{\displaystyle i}
and
j
{\displaystyle j}
where
i
{\displaystyle i}
and
j
{\displaystyle j}
start at 0. Most authors define the Vandermonde matrix as the transpose of the above matrix.
Vandermonde matrices are commonly used in introductory Linear Algebra courses to prove least squares solutions.