| frobenius.matrix {matrixcalc} | R Documentation |
Frobenius Matrix
Description
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 |
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\lbrack {\begin{array}{ccccc}0&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 1}}
\left( {\begin{array}{ccccc}n\\0\end{array}} \right)}\\1&0& \cdots &0&{{{\left( { - 1} \right)}^{n - 2}}
\left( {\begin{array}{ccccc}n\\1\end{array}} \right)}\\0&1& \ddots &0&{{{\left( { - 1} \right)}^{n - 3}}
\left( {\begin{array}{ccccc}n\\2\end{array}} \right)}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\0&0& \cdots &1&{{{\left( { - 1} \right)}^0}
\left( {\begin{array}{ccccc}n\\{n - 1}\end{array}}
\right)}\end{array}}
\right\rbrack.
Value
An order n matrix
Note
If the argument n is not a positive integer that is greater than 1, the function presents an error message and stops.
Author(s)
Frederick Novomestky fnovomes@poly.edu
References
Aceto, L. and D. Trigiante (2001). Matrices of Pascal and Other Greats, American Mathematical Monthly, March 2001, 108(3), 232-245.
Examples
F <- frobenius.matrix( 10 )
print( F )