Given a positive integer n and a ring S, create the complete matrix algebra Mn(S), consisting of all n x n matrices with coefficients in the ring S.
Given a matrix algebra defined as a subalgebra of Mn(S), create the element of R defined by the list L of n2 elements from S.
Given a matrix algebra R defined as a subalgebra of Mn(S) and a sequence Q=[a11, ..., a1n, a21, ..., a2n, ..., an1, ..., ann] of n2 elements of S, return the matrix[ a_11 a_12 ... a_1n ] [ a_21 a_22 ... a_2n ] [ ... ] [ a_n1 a_n2 ... a_nn ]as an element of R. Note that the algebra R must exist before an attempt is made to create matrices.
This function creates a n by n matrix over the finite field K of cardinality q specified in a "Cambridge" format in the general matrix algebra of degree n over K. The parameter t specifies the type of the format. If t is 1, then q is assumed to be less than 10 and the sequence Q must consist of n strings which give the n rows---each string must have length n and contain the entries of that row (each entry is a digit in the range [0, q - 1]). If t is 3 then Q must consist of n2 integers in the range [0, q - 1] which give the entries in row-major order. In either format, if q=pe, where p is prime and e>1, then an entry x is written as a vector using the base-p representation of length e of x and the corresponding element in K is used (see the Finite Fields chapter for details). This function is principally provided for the reading in of large matrices.
Given a monic polynomial p of degree n over a ring R, create the companion matrix C for p as an element of Mn(R). The minimal and characteristic polynomial of C is then p.
If R is a subalgebra of Mn(S) and Q is a sequence of n elements of S, create the diagonal matrix diag( Q[1], Q[2], ..., Q[n] ).
Create the matrix unit E(i, j) in the matrix algebra R, i.e. the matrix having the one of the coefficient ring of R in position (i, j) and zeros elsewhere.
Create a random matrix of the matrix algebra R.
If R is a subalgebra of Mn(S) and t is an element of the ring S, create the scalar matrix t * I in R.
Create the identity matrix In of the matrix algebra R.
Create the zero matrix of the matrix algebra R.
Create the scalar matrix t * I of the matrix algebra R.
Given a commutative ring S and a positive integer n, create the S-algebra R consisting of the n x n matrices over the ring S generated by the elements defined in the list L. Let F denote the algebra Mn(S). Each term Li of the list L must be an expression defining an object of one of the following types:The generators stored for R consist of the elements specified by terms Li together with the stored generators for subalgebras specified by terms of Li. Repetitions of an element and occurrences of scalar matrices are removed.
- (a)
- A sequence of n2 elements of S defining an element of F.
- (b)
- A set or sequence whose terms are sequences of type (a).
- (c)
- An element of F.
- (d)
- A set or sequence whose terms are elements of F.
- (e)
- The null list.
> Q := RationalField(); > A := MatrixAlgebra< Q, 3 | [ 1/3,0,0, 3/2,3,0, -1/2,4,3], > [ 3,0,0, 1/2,-5,0, 8,-1/2,4] >; > A:Maximal; Matrix Algebra of degree 3 with 2 generators over Rational Field Generators: [ 1/3 0 0] [ 3/2 3 0] [-1/2 4 3] [ 3 0 0] [ 1/2 -5 0] [ 8 -1/2 4] > Dimension(A); 6
> K := FiniteField(5); > x := CambridgeMatrix(1, K, 4, [ "1234", "0111", "4321", "1211" ]); > x; [1 2 3 4] [0 1 1 1] [4 3 2 1] [1 2 1 1]
Given a matrix algebra R, construct a structure-constant algebra C isomorphic to R together with the isomorphism from R onto C.
The i-th defining generator for the matrix algebra R.
The coefficient ring S for the matrix algebra R.
Given a matrix algebra R, return the degree n of R.
The set consisting of the defining generators for the matrix algebra R.
The complete matrix algebra Mn(S) in which the matrix algebra R is naturally embedded.
If R is a subring of the matrix algebra Mn(S), then R is considered to act on the free S-module of rank n, consisting of n-tuples over S. The function BaseModule returns this S-module.
The number of defining generators for the matrix algebra R.
Given an element a belonging to the matrix algebra R, return R, i.e. the parent structure for a.
> Q := RationalField(); > A := MatrixAlgebra< Q, 3 | [ 1/3,0,0, 3/2,3,0, -1/2,4,3], > [ 3,0,0, 1/2,-5,0, 8,-1/2,4] >; > CoefficientRing(A); Rational Field > Degree(A); 3 > Ngens(A); 2 > Generators(A); { [ 1/3 0 0] [ 3/2 3 0] [-1/2 4 3], [ 3 0 0] [ 1/2 -5 0] [ 8 -1/2 4] } > Generic(A); Full Matrix Algebra of degree 3 over Rational Field > Dimension(A); 6