This function returns an order n Frobenius matrix that is useful
in numerical mathematics.
Usage
frobenius.matrix(n)
Arguments
n
a positive integer value greater than 1
Value
An order \(n\) matrix
Details
The Frobenius matrix is also called the companion matrix. It arises
in the solution of systems of linear first order differential equations.
The formula for the order \(n\) Frobenius matrix is \({\bf{F}} =
\left[ {\begin{array}{*{20}{c}}0&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 1}}
\left( {\begin{array}{*{20}{c}}n\\0\end{array}} \right)}\\1&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 2}}
\left( {\begin{array}{*{20}{c}}n\\1\end{array}} \right)}\\0&1& \ddots &0&{{{\left( { - 1} \right)}^{n - 3}}
\left( {\begin{array}{*{20}{c}}n\\2\end{array}} \right)}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &1&{{{\left( { - 1} \right)}^0}
\left( {\begin{array}{*{20}{c}}n\\{n - 1}\end{array}}
\right)}\end{array}}
\right]\).
References
Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats,
American Mathematical Monthly, March 2001, 108(3), 232-245.