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.