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Student[LinearAlgebra]

 CharacteristicPolynomial
 construct the characteristic polynomial of a Matrix

 Calling Sequence CharacteristicPolynomial(A, t)

Parameters

 A - Matrix t - name; variable

Description

 • The CharacteristicPolynomial(A, t) command returns the characteristic polynomial in t that has the eigenvalues of Matrix A as its roots (all multiplicities respected).
 This polynomial is the determinant of $\mathrm{Id}t-A$, where Id is the identity Matrix with Dimension(A).

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $M≔⟨⟨1,0,0⟩|⟨1,1,0⟩|⟨0,3,2⟩⟩$
 ${M}{≔}\left[\begin{array}{rrr}{1}& {1}& {0}\\ {0}& {1}& {3}\\ {0}& {0}& {2}\end{array}\right]$ (1)
 > $\mathrm{CharacteristicPolynomial}\left(M,x\right)$
 ${{x}}^{{3}}{-}{4}{}{{x}}^{{2}}{+}{5}{}{x}{-}{2}$ (2)
 > $\mathrm{solve}\left(,x\right)$
 ${2}{,}{1}{,}{1}$ (3)
 > $\mathrm{Eigenvalues}\left(M\right)$
 $\left[\begin{array}{r}{2}\\ {1}\\ {1}\end{array}\right]$ (4)