Construction of Matrix Algebras and their Elements
Construction of the Complete Matrix Algebra
Construction of a Matrix
- elt< R | L >: AlgMat, RngElt -> AlgMatElt
Given a matrix algebra defined as a subalgebra of \(M_n(S)\), create the element of \(R\) defined by the list \(L\) of \(n^2\) elements from \(S\).
- R ! Q: AlgMat, [ RngElt ] -> AlgMatElt
Given a matrix algebra \(R\) defined as a subalgebra of \(M_n(S)\) and a sequence \(Q=[a_{11}, \ldots, a_{1n}, a_{21},\ldots, a_{2n}, \ldots, a_{n1},\ldots, a_{nn}]\) of \(n^2\) elements of \(S\), return the matrix \(\begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix}\)
as an element of \(R\). Note that the algebra \(R\) must exist before an attempt is made to create matrices.
- CambridgeMatrix(t, K, n, Q): RngIntElt, FldFin, RngIntElt, [ ] -> AlgMatElt
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 \(n^2\) integers in the range [0, \(q-1\)] which give the entries in row-major order. In either format, if \(q=p^e\), 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.
- CompanionMatrix(p): RngUPolElt -> AlgMatElt
Given a monic polynomial \(p\) of degree \(n\) over a ring \(R\), create the companion matrix \(C\) for \(p\) as an element of \(M_n(R)\). The minimal and characteristic polynomial of \(C\) is then \(p\).
- DiagonalMatrix(R, Q): AlgMat, [ RngElt ] -> AlgMatElt
If \(R\) is a subalgebra of \(M_n(S)\) and \(Q\) is a sequence of \(n\) elements of \(S\), create the diagonal matrix \(diag( Q[1], Q[2], \ldots, Q[n] )\).
- MatrixUnit(R, i, j): AlgMat, RngIntElt, RngIntElt -> AlgMatElt
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.
- ScalarMatrix(R, t): AlgMat, RngElt -> AlgMatElt
If \(R\) is a subalgebra of \(M_n(S)\) and \(t\) is an element of the ring \(S\), create the scalar matrix \(t*I\) in \(R\).
Constructing a General Matrix Algebra
- MatrixAlgebra<S, n | L>: Rng, RngIntElt, List -> AlgMat
- MatrixRing<S, n | L>: Rng, RngIntElt, List -> AlgMat
Given a commutative ring \(S\) and a positive integer \(n\), create the \(S\)-algebra \(R\) consisting of the \(n \times n\) matrices over the ring \(S\) generated by the elements defined in the list \(L\). Let \(F\) denote the algebra \(M_n(S)\). Each term \(L_i\) of the list \(L\) must be an expression defining an object of one of the following types:
- (a)
A sequence of \(n^2\) 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.
The generators stored for \(R\) consist of the elements specified by terms \(L_i\) together with the stored generators for subalgebras specified by terms of \(L_i\). Repetitions of an element and occurrences of scalar matrices are removed.
- Example: Algebra of Lower-Triangular Matrices (ex-0b78f0)
Run in calculator (opens in a new tab)
We demonstrate the use of the matrix algebra constructor by creating an algebra of \(3 \times 3\) lower-triangular matrices over the rational field.
> 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
The Invariants of a Matrix Algebra
- BaseRing(R): AlgMatV -> Rng
- CoefficientRing(R): AlgMatV -> Rng
The coefficient ring \(S\) for the matrix algebra \(R\).
- Generators(R): AlgMat -> { AlgMatElt}
The set consisting of the defining generators for the matrix algebra \(R\).
- Generic(R): AlgMat -> AlgMat
The complete matrix algebra \(M_n(S)\) in which the matrix algebra \(R\) is naturally embedded.
- BaseModule(R): AlgMatV -> ModTup
If \(R\) is a subring of the matrix algebra \(M_n(S)\), then \(R\) is considered to act on the free \(S\)-module of rank \(n\), consisting of \(n\)-tuples over \(S\). The function
BaseModulereturns this \(S\)-module.
- NumberOfGenerators(R): AlgMat -> { AlgMatElt}
- Ngens(R): AlgMat -> { AlgMatElt}
The number of defining generators for the matrix algebra \(R\).
- Parent(a): AlgMatElt -> AlgMat
Given an element \(a\) belonging to the matrix algebra \(R\), return \(R\), i.e. the parent structure for \(a\).
- Example: Invariants of a Lower-Triangular Algebra (ex-97af59)
Run in calculator (opens in a new tab)
We illustrate the use of these functions by applying them to the algebra of \(3 \times 3\) lower-triangular matrices over the rational field constructed above.
> 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