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Student[LinearAlgebra]

 MinimalPolynomial
 construct the minimal polynomial of a Matrix

 Calling Sequence MinimalPolynomial(A, t)

Parameters

 A - Matrix t - name; variable

Description

 • The MinimalPolynomial(A, t) command returns a polynomial in t 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.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $A≔⟨⟨3,0,1⟩|⟨-1,2,-1⟩|⟨0,0,2⟩⟩$
 ${A}{:=}\left[\begin{array}{rrr}{3}& {-}{1}& {0}\\ {0}& {2}& {0}\\ {1}& {-}{1}& {2}\end{array}\right]$ (1)
 > $\mathrm{mp}≔\mathrm{MinimalPolynomial}\left(A,x\right)$
 ${\mathrm{mp}}{:=}{{x}}^{{2}}{-}{5}{}{x}{+}{6}$ (2)
 > $\mathrm{divide}\left(\mathrm{CharacteristicPolynomial}\left(A,x\right),\mathrm{mp}\right)$
 ${\mathrm{true}}$ (3)
 > $P≔\mathrm{unapply}\left(\mathrm{mp},x\right)$
 ${P}{:=}{x}{→}{{x}}^{{2}}{-}{5}{}{x}{+}{6}$ (4)
 > $P\left(A\right)$
 $\left[\begin{array}{rrr}{0}& {0}& {0}\\ {0}& {0}& {0}\\ {0}& {0}& {0}\end{array}\right]$ (5)