The JordanForm(A) function returns the Jordan form J of Matrix A.

The Jordan form J has the structure of having Jordan block submatrices along its diagonal. The diagonal entries of these Jordan blocks are the eigenvalues of A (and also of J).

The Jordan form is unique up to permutations of the Jordan blocks.

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 J (the Jordan form) or Q (the transition 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 $Q^{\mathrm{-1}}·A·Q=J$.

The constructor options provide additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix constructor that builds the result. 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.

This function is part of the LinearAlgebra package, and so it can be used in the form JordanForm(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[JordanForm](..).

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

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

$J≔\mathrm{JordanForm}\left(A\right)$

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

$\mathrm{Eigenvalues}\left(A\,\mathrm{output}='\mathrm{list}'\right)$

$\left[0\,0\,2\,2\right]$

$Q≔\mathrm{JordanForm}\left(A\,\mathrm{output}='Q'\right)$

$Q≔\left[\begin{array}{cccc}\mathrm{-1}& -\frac{3}{2}& 2& 1\\ \mathrm{-1}& -\frac{1}{2}& \frac{1}{2}& 0\\ \mathrm{-1}& \frac{1}{2}& \frac{1}{2}& 0\\ \mathrm{-1}& -\frac{3}{2}& \frac{5}{2}& 1\end{array}\right]$

$Q^{-1}·A·Q$

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