Subsections

• Arithmetic with Matrices
• Predicates for Matrices
• Matrix Invariants
• Coercion between Matrix Structures

Product of matrix g and matrix h, where g and h belong to the same generic group U. If g and h both belong to the same proper subgroup G of U, then the result will be returned as an element of G; if g and h belong to subgroups H and K of a subgroup G of U then the product is returned as an element of G. Otherwise, the product is returned as an element of U.

The n-th power of the matrix g, where n is a positive or negative integer.

Product of the matrix g by the inverse of the matrix h, i.e. the element g * h^(-1). Here g and h must belong to the same generic group U. The rules for determining the parent group of g / h are the same as for g * h.

Conjugate of the matrix g by the matrix h, i.e. the element h^(-1) * g * h. Here g and h must belong to the same generic group U. The rules for determining the parent group of g^h are the same as for g * h.

Commutator of the matrices g and h, i.e. the element g^(-1) * h^(-1) * g * h. Here g and h must belong to the same generic group U. The rules for determining the parent group of (g, h) are the same as those for g * h.

Given r matrices g_1, ..., g_r belonging to a common group, return their commutator. Commutators are left-normed, so they are evaluated from left to right.

> K<w> := FiniteField(4);
> GL34 := GeneralLinearGroup(3, K);
> x := GL34 ! [1,w,0, 0,w,1, w^2,0,1];
> y := GL34 ! [1,0,0, 1,w,0, 1,1,w];
> x;
[  1   w   0]
[  0   w   1]
[w^2   0   1]
> y;
[1 0 0]
[1 w 0]
[1 1 w]
> x*y;
[w^2 w^2   0]
[w^2   w   w]
[  w   1   w]
> x^10;
[  w   w   1]
[  w   1   1]
[  w w^2   w]
> x^-1;
[w^2 w^2 w^2]
[  1   w   w]
[  w   w w^2]
> x^y;
[w^2 w^2   0]
[  0 w^2   1]
[w^2 w^2   w]
> x/y;
[  0   1   0]
[  0 w^2 w^2]
[  w   w w^2]
> (x, y);
[  0   w   w]
[  w w^2   1]
[w^2   w w^2]
> (x,y,y);
[w^2   w w^2]
[w^2   w   0]
[w^2   1   w]

g eq h : GrpMatElt, GrpMatElt -> BoolElt

Given matrices g and h belonging to the same generic group, return true if g and h are the same element, false otherwise.

Given matrices g and h belonging to the same generic group, return true if g and h are distinct elements, false otherwise.

IsId(g) : GrpMatElt -> BoolElt

True if the matrix g is the identity matrix.

True if the matrix g is a scalar matrix.

Given a matrix g of the group G, return the i-th row of g as an element of the natural module of G.

g[i, j] : GrpMatElt, RngIntElt, RngIntElt -> RngElt

Given a matrix g of the group G, return the (i, j)-th entry of g as an element of the coefficient ring of G.

NumberOfRows(g) : GrpMatElt -> RngIntElt

Nrows(g) : GrpMatElt -> RngIntElt

NumberOfColumns(g) : GrpMatElt -> RngIntElt

Ncols(g) : GrpMatElt -> RngIntElt

The degree of the matrix g, i.e. the number of rows/columns of g.

True iff the matrix g has finite order. The function rigorously proves its result (i.e., the result is not probable).

Order of the matrix g. If g has infinite order, an error ensues.

The order of the matrix g returned as a factorization sequence. It is more efficient to use this function than to factorize the result given by Order(g). If g has infinite order, an error ensues.

The determinant of the matrix g.

The trace of the matrix g.

The transpose of the matrix g returned as an element of the general linear group containing g.

    Al: MonStg                          Default: "Modular"

    Proof: BoolElt                      Default: true

Given a matrix g belonging to a subgroup of GL(n, R), where R is a field of Euclidean Domain, return the characteristic polynomial of g as an element of the univariate polynomial ring over R. For details on the parameters, see the function CharacteristicPolynomial in the Matrix Algebra chapter.

Given a matrix g belonging to a subgroup of GL(n, R), where R is a field or Z, return the minimal polynomial of g as an element of the univariate polynomial ring over R.

Given a matrix g belonging to a subgroup of GL(n, R), where R has a polynomial roots algorithm, return the eigenvalues of g as a set of pairs consisting of an eigenvalue of g and its multiplicity.

Given a matrix g belonging to a subgroup G of GL(n, R), where R is a field or Euclidean Domain, return the eigenspace of g corresponding to the eigenvalue a. The eigenspace is returned as a subspace of the natural module for G.

Given a matrix g belonging to a subgroup G of GL(n, K), where K is a field, return the rational form of g. For each block other than the last block, the polynomial corresponding to that block divides the polynomial corresponding to the next block. This function returns three values:

• The rational canonical form f of g, returned as an element of the full matrix algebra corresponding to G;
• A matrix t such that t * g * t^(-1) = f;
• A sequence containing the polynomials corresponding to each block (each non-last one dividing the next).

Given a matrix g belonging to a subgroup G of GL(n, K), where K is a field, return the primary rational canonical form p of g. Each block of p corresponds to a power of an irreducible polynomial. This function returns three values:

• The primary rational canonical form p of g, returned as an element of the full matrix algebra corresponding to G;
• A matrix t such that t * g * t^(-1) = p;
• A sequence of pairs corresponding to the blocks of p where each pair consists of the irreducible polynomial and multiplicity making up the block.

Given a matrix g belonging to a subgroup G of GL(n, K), where K is a field, return the (generalized) Jordan canonical form of g. This function returns three values:

• The Jordan canonical form j of g, returned as an element of the full matrix algebra corresponding to G;
• A matrix t such that t * g * t^(-1) = j;
• A sequence of pairs corresponding to the blocks of j where each pair consists of the irreducible polynomial and multiplicity making up the block.

The invariant factors of the matrix g belonging to the matrix group G. This is the same as the third return value of RationalForm(g).

Given a matrix g belonging to a subgroup G of GL(n, K), where K is a field, return the primary invariant factors of g. This is the same as the third value returned by PrimaryRationalForm(g) or JordanForm(g).

> K<w> := FiniteField(4);
> GL34 := GeneralLinearGroup(3, K);
> x := GL34 ! [w,0,1, 0,1,0, 1,0,1];
> x;
[w 0 1]
[0 1 0]
[1 0 1]
> Degree(x);
3
> Determinant(x);
w^2
> Trace(x);
w
> Order(x);
15
> m<t> := MinimalPolynomial(x);
> m;
t^3 + w*t^2 + w^2
> Factorization(m);
[
    <t + 1, 1>,
    <t^2 + w^2*t + w^2, 1>
]
> c<t> := CharacteristicPolynomial(x);
> c;
t^3 + w*t^2 + w^2

A square non-singular matrix may be defined as an element of any of the following structures:

• A subring of the complete matrix ring M_(n)(R);
• A subgroup of the general linear group GL(n, R);
• A submodule of the matrix module M^((m x n))(R).

Transfer the matrix g from a group into a matrix ring R.

G ! r : GrpMat, RngMatElt -> GrpMatElt

Transfer the matrix r from a ring into a matrix group G.

M ! g : ModMat, GrpMatElt -> ModMatElt

Transfer the matrix g from a group into a matrix module M.

G ! m : GrpMat, ModMatElt -> GrpMatElt

Transfer the matrix m from a module into a matrix group G.