The MinimalPolynomial(A, lambda) function returns a polynomial in lambda that is the minimal polynomial of Matrix A.

The minimal polynomial of A is the polynomial of lowest degree that divides every polynomial which has A as a zero.

The minimal polynomial of A always divides the characteristic polynomial of A.

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

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

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

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

$\mathrm{mp}≔\mathrm{MinimalPolynomial}\left(A\,x\right)$

$\mathrm{mp}≔x^2-5x+6$

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

$\mathrm{true}$

$P≔\mathrm{unapply}\left(\mathrm{mp}\,x\right)$

$P≔x\mapsto x^2-5\cdot x+6$

$P\left(A\right)$

$\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$