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The functions in this section deal with minimal and characteristic
polynomials.
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the
minimal polynomial of A. This is defined to be the unique monic univariate
polynomial f(x) of minimal degree such that f(A) = 0, and f(x)
always divides the characteristic polynomial of A. The coefficient ring
R is currently restricted to being a field or the integer ring Z.
Setting the parameter Proof to false suppresses proof of correctness.
Al: MonStg Default: "Modular"
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the
characteristic polynomial of A. This is defined to be the monic univariate
polynomial Det(x - A)∈R[x] where R[x] is the univariate polynomial
ring over R. R may be any commutative ring.
The parameter Al allows the user to specify which algorithm that
is to be employed. The
algorithm "Modular" (the default) may be used for matrices over Z
and Q---in such a case the parameter Proof can also be used to
suppress proof of correctness.
The algorithm "Hessenberg", available
for matrices over fields, works by first reducing the matrix to
Hessenberg form.
The algorithm "Interpolation", available for
matrices over Z and Q, works by evaluating the characteristic
matrix of a at various points and then interpolating.
The algorithm "Trace", available for matrices over fields, works by
calculating the traces of powers of a.
Since V2.8, none of these algorithms are now recommended for matrices
over Z or Q, as the
new p-adic modular algorithm over the integers is extremely fast.
MCPolynomials(A) : Mtrx -> RngUPolElt, RngUPolElt
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the
minimal and characteristic polynomials of A.
For some rings,
both polynomials can be computed at the same time, so in such cases
it will be more efficient to use this function than to call
MinimalPolynomial and CharacteristicPolynomials separately.
Setting the parameter Proof to false suppresses proof of correctness.
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the factorization of
the minimal polynomial of A. This is equivalent to Factorization(MinimalPolynomial(A)), but may be faster than that for
some coefficient rings (in particular, Z and Q).
Setting the parameter Proof to false suppresses proof of correctness.
Al: MonStg Default: "Modular"
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the factorisation of
the characteristic polynomial of A. This function returns the same result
as the command
Factorisation(CharacteristicPolynomial(A)), but may be faster than
that for some coefficient rings (in particular, Z and Q). The
parameters are as for the function CharacteristicPolynomial above
(setting the parameter Proof to false suppresses proof of
correctness).
FactoredMCPolynomials(A: parameters) : Mtrx -> [<RngUPolElt, RngIntElt>], [<RngUPolElt, RngIntElt>]
Al: MonStg Default: "Modular"
Proof: BoolElt Default: true
Given a square matrix A over a ring R, return the factorizations of
the minimal and characteristic polynomials of A, respectively.
For some rings, both polynomials can be computed and factored at the same
time, so in such cases it will be more efficient to use this
function than to call the above functions separately.
Setting the parameter Proof to false suppresses proof of correctness.
Given a square matrix A over a ring R, return the
eigenvalues of A as a set of pairs, each of
which gives the value of a distinct eigenvalue and its multiplicity.
Factorization of polynomials over the base ring R must be
possible.
Given a square matrix A over a ring R, and an element e
of R, return the eigenspace of A corresponding to e, which
is Nullspace(A - e).
If the ring element e is not a eigenvalue for the matrix A
then the trivial nullspace is returned.
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