The FrobeniusForm(A) command returns the Frobenius form F of square Matrix A.

This function can also be invoked using the RationalCanonicalForm command.

The Frobenius form Matrix F has the following structure:

where the $C_i$ are companion Matrices associated with polynomials $p_1,p_2,..,p_k$ where the $p_i$ are a factorization of the characteristic polynomial of A with the property that $p_i$ divides $p_{i-1}$, for $i$ = 2..k.

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

The columns of Q form a rational canonical basis for A.

The output option (out) determines the content of the returned expression sequence.

Depending on what is included in the output option, an expression sequence containing one or more of the factors F (the Frobenius form), or Q (the transformation Matrix) can be returned. If output is a list, the objects are returned in the same order as specified in the list.

The returned Matrix objects have the property that $\mathrm{MatrixInverse}\left(Q\right)·A·Q=F$.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result(s). These options may also be provided in the form outputoptions[o]=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions[o] option, the latter takes precedence (regardless of the order).

The following list indicates permissible values for index [o] of outputoptions with their corresponding meaning.

$\mathrm{with}\left(\mathrm{LinearAlgebra}\right)\:$

$A≔⟨⟨0\,1\,1\,1\,1⟩\left|⟨2\,-2\,0\,-2\,-4⟩\right|⟨0\,0\,1\,1\,3⟩\left|⟨-6\,0\,-3\,-1\,-3⟩\right|⟨2\,2\,2\,2\,4⟩⟩$

$A≔\left[\begin{array}{ccccc}0& 2& 0& \mathrm{-6}& 2\\ 1& \mathrm{-2}& 0& 0& 2\\ 1& 0& 1& \mathrm{-3}& 2\\ 1& \mathrm{-2}& 1& \mathrm{-1}& 2\\ 1& \mathrm{-4}& 3& \mathrm{-3}& 4\end{array}\right]$

$\mathrm{FrobeniusForm}\left(A\right)$

$\left[\begin{array}{ccccc}0& 0& 4& 0& 0\\ 1& 0& \mathrm{-2}& 0& 0\\ 0& 1& 2& 0& 0\\ 0& 0& 0& 0& \mathrm{-2}\\ 0& 0& 0& 1& 0\end{array}\right]$

$\mathrm{factor}\left(\mathrm{CharacteristicPolynomial}\left(A\,x\right)\right)$

$\left(x-2\right){\left(x^2+2\right)}^2$

$M≔\mathrm{BandMatrix}\left(\left[\left[2\,2\,2\,2\right]\,\left[1\,1\right]\right]\,0\right)$

$M≔\left[\begin{array}{cccc}2& 1& 0& 0\\ 0& 2& 1& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 2\end{array}\right]$

$F,Q≔\mathrm{FrobeniusForm}\left(M\,\mathrm{output}=\left['F'\,'Q'\right]\right)$

$F,Q≔\left[\begin{array}{cccc}0& 0& 8& 0\\ 1& 0& \mathrm{-12}& 0\\ 0& 1& 6& 0\\ 0& 0& 0& 2\end{array}\right],\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 1& 4& 0\\ 1& 2& 4& 0\\ 1& 2& 4& 1\end{array}\right]$

$Q^{-1}·M·Q$

$\left[\begin{array}{cccc}0& 0& 8& 0\\ 1& 0& \mathrm{-12}& 0\\ 0& 1& 6& 0\\ 0& 0& 0& 2\end{array}\right]$