The (indexed) parameter R, which specifies the domain of computation, a commutative ring, must be a Maple table/module which has the following values/exports:

R[`0`] : a constant for the zero of the ring R

R[`1`] : a constant for the (multiplicative) identity of R

R[`+`] : a procedure for adding elements of R (nary)

R[`-`] : a procedure for negating and subtracting elements of R (unary and binary)

R[`*`] : a procedure for multiplying elements of R (binary and commutative)

R[`=`] : a boolean procedure for testing if two elements of R are equal

A must be a square (n x n) Matrix of values from R.

The optional parameter x must be a name.

CharacteristicPolynomial[R](A) returns a Vector V of dimension n+1 of values in R containing the coefficients of the characteristic polynomial of A. The characteristic polynomial is the polynomial V[1]*x^n + V[2]*x^(n-1) + ... + V[n]*x + V[n+1].

CharacteristicPolynomial[R](A,x) returns the characteristic polynomial as a Maple expression in the variable x. This option should only be used if the data type for R is compatible with Maple's * operator. For example, if the elements of R are represented by Vectors, or Arrays, then this option should not be used because Vector([1,2,3])*x is simplified to Vector([x,2*x,3*x]).

The optional argument output=... specifies the form of the output. In the case output=expanded, the characteristic polynomial is returned as one Vector encoding the characteristic polynomial in expanded form. In the case output=factored, the characteristic polynomial is returned as a sequence of the form m, [v1, v2, ...] where m is a non-negative integer, and v1, v2, ... are Vectors of elements of R representing (not necessarily irreducible) factors of the characteristic polynomial. The integer m represents the factor x^m. The implementation looks for diagonal blocks and computes the characteristic polynomial of each block separately.

The optional argument method=... specifies the algorithm to be used. The option method=Berkowitz directs the code to use the Berkowitz algorithm, which uses O(n^4) arithmetic operations in R. The option method=Hessenberg directs the code to use the Hessenberg algorithm, which uses O(n^3) arithmetic operations in R but requires R to be a field, i.e., the following operation must be defined:

R[`/`]: a procedure for dividing two elements of R

If method=... is not given, and the operation R[`/`] is defined, then the Hessenberg algorithm is used, otherwise the Berkowitz algorithm is used.

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

$Z\left[\mathrm{`0`}\right],Z\left[\mathrm{`1`}\right],Z\left[\mathrm{`+`}\right],Z\left[\mathrm{`-`}\right],Z\left[\mathrm{`*`}\right],Z\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`=`}\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,1\,4\right]\,\left[3\,2\,1\right]\,\left[0\,0\,5\right]\right]\right)$

$A≔\left[\begin{array}{ccc}2& 1& 4\\ 3& 2& 1\\ 0& 0& 5\end{array}\right]$

$C≔\mathrm{CharacteristicPolynomial}\left[Z\right]\left(A\right)$

$C≔\left[\begin{array}{c}1\\ \mathrm{-9}\\ 21\\ \mathrm{-5}\end{array}\right]$

$⟨x^3\left|x^2\right|x|1⟩·C$

$x^3-9x^2+21x-5$

$C≔\mathrm{CharacteristicPolynomial}\left[Z\right]\left(A\,x\right)$

$C≔x^3-9x^2+21x-5$

$n,C≔\mathrm{CharacteristicPolynomial}\left[Z\right]\left(A\,\mathrm{output}=\mathrm{factored}\right)$

$n,C≔0,\left[\left[\begin{array}{c}1\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}1\\ \mathrm{-4}\\ 1\end{array}\right]\right]$

$x^n\left(x+C\left[1\right]\left[2\right]\right)\left(x^2+C\left[2\right]\left[2\right]x+C\left[2\right]\left[3\right]\right)$

$\left(x-5\right)\left(x^2-4x+1\right)$

$\mathrm{CharacteristicPolynomial}\left[Z\right]\left(A\,x\,\mathrm{output}=\mathrm{factored}\right)$

$\left(x-5\right)\left(x^2-4x+1\right)$

$Q\left[\mathrm{`0`}\right],Q\left[\mathrm{`1`}\right],Q\left[\mathrm{`+`}\right],Q\left[\mathrm{`-`}\right],Q\left[\mathrm{`*`}\right],Q\left[\mathrm{`/`}\right],Q\left[\mathrm{`=`}\right]≔0,1,\mathrm{`+`},\mathrm{`-`},\mathrm{`*`},\mathrm{`/`},\mathrm{`=`}\:$

$A≔\mathrm{Matrix}\left(\left[\left[2\,-7\,-3\,4\right]\,\left[1\,-3\,-4\,5\right]\,\left[0\,0\,5\,-7\right]\,\left[0\,0\,0\,0\right]\right]\right)$

$A≔\left[\begin{array}{cccc}2& \mathrm{-7}& \mathrm{-3}& 4\\ 1& \mathrm{-3}& \mathrm{-4}& 5\\ 0& 0& 5& \mathrm{-7}\\ 0& 0& 0& 0\end{array}\right]$

$n,C≔\mathrm{CharacteristicPolynomial}\left[Q\right]\left(A\,\mathrm{output}=\mathrm{factored}\right)$

$n,C≔1,\left[\left[\begin{array}{c}1\\ \mathrm{-5}\end{array}\right]\,\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]\right]$

$x^n\left(x+C\left[1\right]\left[2\right]\right)\left(x^2+C\left[2\right]\left[2\right]x+C\left[2\right]\left[3\right]\right)$

$x\left(x-5\right)\left(x^2+x+1\right)$

$C≔\mathrm{CharacteristicPolynomial}\left[Q\right]\left(A\right)$

$C≔\left[\begin{array}{c}1\\ \mathrm{-4}\\ \mathrm{-4}\\ \mathrm{-5}\\ 0\end{array}\right]$

$⟨x^4\left|x^3\right|x^2\left|x\right|1⟩·C$

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