Important: The linalg package has been deprecated. Use the superseding command, LinearAlgebra[FrobeniusForm], instead.

- For information on migrating linalg code to the new packages, see examples/LinearAlgebraMigration.

The function frobenius(A) or ratform(A) computes and returns the Frobenius form F of a matrix A. This is also known as the Rational Canonical form of a matrix.

F has the following structure: F = diag(C[1], C[2],.., C[k]) where the $C_i$ are companion matrices associated with polynomials $p_1,p_2,..,p_k$ with the property that $p_i$ divides $p_{i-1}$, for $i$ = 2..k.

If called in the form frobenius(A, 'P'), then P will be assigned the transformation matrix corresponding to the Frobenius form, that is, the matrix P such that $\mathrm{inverse}\left(P\right)AP=F$.

The Frobenius form defined in this way is unique (if we require that $p_i$ divides $p_{i-1}$).

If the sizes of the blocks $C_1,C_2,...,C_k$ are $n_1,n_2,...,n_k$ respectively, then the columns of the matrix P are the vectors $\[f_1,A{f}_1,\mathrm{..},A^{n_1-1}{f}_1,f_2,...,f_k,A{f}_k,...,A^{n_k-1}{f}_k\]$ 

These functions are part of the linalg package, and so can be used in the form frobenius(..) or ratform(..) only after performing the command with(linalg), or with(linalg,frobenius) or with(linalg,ratform). These functions can always be accessed in the long form linalg[frobenius](..) or linalg[ratform](..).

$\mathrm{with}\left(\mathrm{linalg}\,\mathrm{frobenius}\right)\:$

$A≔\mathrm{array}\left(\left[\left[-9\,21\,-15\,4\,2\,0\right]\,\left[-10\,21\,-14\,4\,2\,0\right]\,\left[-8\,16\,-11\,4\,2\,0\right]\,\left[-6\,12\,-9\,3\,3\,0\right]\,\left[-4\,8\,-6\,0\,5\,0\right]\,\left[-2\,4\,-3\,0\,1\,3\right]\right]\right)$

$A≔\left[\begin{array}{cccccc}\mathrm{-9}& 21& \mathrm{-15}& 4& 2& 0\\ \mathrm{-10}& 21& \mathrm{-14}& 4& 2& 0\\ \mathrm{-8}& 16& \mathrm{-11}& 4& 2& 0\\ \mathrm{-6}& 12& \mathrm{-9}& 3& 3& 0\\ \mathrm{-4}& 8& \mathrm{-6}& 0& 5& 0\\ \mathrm{-2}& 4& \mathrm{-3}& 0& 1& 3\end{array}\right]$

$\mathrm{frobenius}\left(A\,P\right)$

$\left[\begin{array}{cccccc}0& 0& 0& 0& 15& 0\\ 1& 0& 0& 0& \mathrm{-47}& 0\\ 0& 1& 0& 0& 56& 0\\ 0& 0& 1& 0& \mathrm{-32}& 0\\ 0& 0& 0& 1& 9& 0\\ 0& 0& 0& 0& 0& 3\end{array}\right]$

$\mathrm{print}\left(P\right)$

$\left[\begin{array}{cccccc}\frac{2175}{53}& \frac{37329}{265}& \frac{120051}{265}& \frac{308409}{265}& \frac{644331}{265}& 0\\ \frac{2511}{53}& \frac{7533}{53}& \frac{22599}{53}& \frac{55701}{53}& \frac{111999}{53}& 0\\ \frac{11211}{265}& \frac{30273}{265}& \frac{90147}{265}& \frac{224073}{265}& \frac{456507}{265}& 0\\ \frac{7347}{265}& \frac{22041}{265}& \frac{68139}{265}& \frac{172161}{265}& \frac{357219}{265}& 0\\ \frac{5163}{265}& \frac{15489}{265}& \frac{47811}{265}& \frac{121929}{265}& \frac{259611}{265}& 0\\ \frac{2714}{265}& \frac{8142}{265}& \frac{25098}{265}& \frac{64542}{265}& \frac{140538}{265}& 1\end{array}\right]$

Ozello, Patrick. Calcul Exact des Formes de Jordan et de Frobenius d'une Matrice. PhD Thesis, Joseph Fourier University, Grenoble, France, 1987.

Martin, K., and Olazabal, J.M. "An Algorithm to Compute the Change Basis for the Rational Form of K-endomorphisms." Extracta Mathematicae, (August 1991): 142-144.