The CharacteristicPolynomial(A, lambda) function returns the characteristic polynomial in lambda that has the eigenvalues of Matrix A as its roots (all multiplicities respected).

This polynomial is the determinant of $\mathrm{I}\mathrm{\lambda }-A$, where I is the identity Matrix with dimension(A).

When the input is an integer Matrix, the LinearAlgebra:-Modular:-IntegerCharacteristicPolynomial routine is used. 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 function is part of the LinearAlgebra package, and so it can be used in the form CharacteristicPolynomial(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[CharacteristicPolynomial](..).

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

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

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

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

$x^3-4x^2+5x-2$

$\mathrm{solve}\left(\,x\right)$

$2,1,1$

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

$\left[2\,1\,1\right]$

Abdeljaoued, J. "The Berkowitz Algorithm, Maple and Computing the Characteristic Polynomial in an Arbitrary Commutative Ring." MapleTech Vol. 4 No. 3. Birkhauser, (1997): 21-32.

De Boor, C. "An Empty Exercise." ACM SIGNUM Newsletter, Vol. 25 No. 2. (1990): 2-6.