The IntegerCharacteristicPolynomial function computes the characteristic polynomial for a square Matrix with integer entries. This is a programmer level function, and it does not perform argument checking. Thus, argument checking must be handled external to this function.

Note: The IntegerCharacteristicPolynomial routine uses a probabilistic approach that achieves great gains for structured systems. Information on controlling the probabilistic behavior can be found in _EnvProbabilistic.

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form IntegerCharacteristicPolynomial(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][IntegerCharacteristicPolynomial](..).

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

$M≔\mathrm{Matrix}\left(\left[\left[2\,1\,3\right]\,\left[4\,3\,1\right]\,\left[-2\,1\,-3\right]\right]\right)$

$M≔\left[\begin{array}{ccc}2& 1& 3\\ 4& 3& 1\\ \mathrm{-2}& 1& \mathrm{-3}\end{array}\right]$

$\mathrm{IntegerCharacteristicPolynomial}\left(M\,x\right)$

$x^3-2x^2-8x-20$

$\mathrm{LinearAlgebra}:-\mathrm{CharacteristicPolynomial}\left(M\,x\right)$

$x^3-2x^2-8x-20$

$M≔\mathrm{LinearAlgebra}:-\mathrm{RandomMatrix}\left(50\,50\,\mathrm{density}=0.5\,\mathrm{generator}=-99..99\,\mathrm{outputoptions}=\left[\mathrm{datatype}=\mathrm{integer}\right]\right)$

$M≔\begin{array}{c}\left[\begin{array}{ccccccccccc}39& \mathrm{-64}& 0& 0& 0& 19& \mathrm{-16}& 0& 50& \mathrm{-3}& \dots \\ \mathrm{-87}& 48& 0& 35& \mathrm{-20}& \mathrm{-75}& 0& 0& 0& 0& \dots \\ \mathrm{-48}& 0& \mathrm{-46}& \mathrm{-1}& 0& 0& 81& \mathrm{-1}& \mathrm{-5}& 0& \dots \\ 0& 0& 56& \mathrm{-24}& 0& 81& \mathrm{-78}& 26& 0& \mathrm{-74}& \dots \\ \mathrm{-62}& 7& 73& 2& 13& 0& 41& \mathrm{-4}& \mathrm{-21}& 90& \dots \\ \mathrm{-63}& 67& 0& 0& 0& 0& 0& 12& \mathrm{-86}& 0& \dots \\ 0& 0& 94& \mathrm{-98}& 0& 0& 5& \mathrm{-60}& 0& 58& \dots \\ \mathrm{-97}& 18& 0& 0& 30& \mathrm{-57}& 0& 82& 43& 0& \dots \\ 0& 0& \mathrm{-71}& 70& 0& \mathrm{-54}& 0& 0& \mathrm{-20}& 0& \dots \\ \mathrm{-57}& 45& \mathrm{-10}& 0& 0& \mathrm{-46}& \mathrm{-80}& \mathrm{-95}& 0& 0& \dots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& \end{array}\right]\\ \hfill \text{50 × 50 Matrix}\end{array}$

$\mathrm{tt}≔\mathrm{time}\left(\right)\:$

$\mathrm{p1}≔\mathrm{IntegerCharacteristicPolynomial}\left(M\,x\right)\:$

$\mathrm{t1}≔\mathrm{time}\left(\right)-\mathrm{tt}$

$\mathrm{t1}≔0.028$

$\mathrm{\_EnvDisableModular}≔\mathrm{true}\:$

$\mathrm{tt}≔\mathrm{time}\left(\right)\:$

$\mathrm{p2}≔\mathrm{LinearAlgebra}:-\mathrm{CharacteristicPolynomial}\left(M\,x\right)\:$

$\mathrm{t2}≔\mathrm{time}\left(\right)-\mathrm{tt}$

$\mathrm{t2}≔0.757$

$\mathrm{expand}\left(\mathrm{p1}-\mathrm{p2}\right)$

$0$

$\frac{\mathrm{t2}}{\mathrm{t1}}$

$27.03571429$