MATRIX ALGEBRAS
Acknowledgements
Introduction
Construction of Matrix Algebras and their Elements
Construction of the Complete Matrix Algebra
Construction of a Matrix
Constructing a General Matrix Algebra
The Invariants of a Matrix Algebra
Construction of Subalgebras, Ideals and Quotient Rings
The Construction of Extensions and their Elements
The Construction of Direct Sums and Tensor Products
Construction of Direct Sums and Tensor Products of Elements
Operations on Matrix Algebras
Changing Rings
Elementary Operations on Elements
Arithmetic
Predicates
Comparison
Properties of Elements
Elements of Mn as Homomorphisms
Elementary Operations on Subalgebras and Ideals
Bases
Intersection of Subalgebras
Membership and Equality
Accessing and Modifying a Matrix
Indexing
Extracting and Inserting Blocks
Joining Matrices
Row and Column Operations
Canonical Forms
Canonical Forms for Matrices over Euclidean Domains
Canonical Forms for Matrices over a Field
Diagonalising Commutative Algebras over a Field
Solutions of Systems of Linear Equations
Presentations for Matrix Algebras
Quotients and Idempotents
Generators and Presentations
Solving the Word Problem
Bibliography
Introduction
Construction of Matrix Algebras and their Elements
Construction of the Complete Matrix Algebra
MatrixAlgebra(S, n) : Rng, RngIntElt -> AlgMat
Construction of a Matrix
elt< R | L > : AlgMat, RngElt -> AlgMatElt
R ! Q : AlgMat, [ RngElt ] -> AlgMatElt
CambridgeMatrix(t, K, n, Q) : RngIntElt, FldFin, RngIntElt, [ ] -> AlgMatElt
CompanionMatrix(p) : RngUPolElt -> AlgMatElt
DiagonalMatrix(R, Q) : AlgMat, [ RngElt ] -> AlgMatElt
MatrixUnit(R, i, j) : AlgMat, RngIntElt, RngIntElt -> AlgMatElt
Random(R) : AlgMat -> AlgMatElt
ScalarMatrix(R, t) : AlgMat, RngElt -> AlgMatElt
R ! 1 : AlgMat, RngIntElt -> AlgMatElt
R ! 0 : AlgMat, RngIntElt -> AlgMatElt
R ! t : AlgMat, RngIntElt -> AlgMatElt
Constructing a General Matrix Algebra
MatrixAlgebra<S, n | L> : Rng, RngIntElt, List -> AlgMat
Example AlgMat_Creation (H92E1)
Example AlgMat_Cambridge (H92E2)
Algebra(R) : AlgMatV -> AlgGen, Map
The Invariants of a Matrix Algebra
R . i : AlgMat, RngIntElt -> AlgMatElt
BaseRing(R) : AlgMatV -> Rng
Degree(R) : AlgMatV -> RngIntElt
Generators(R) : AlgMat -> { AlgMatElt }
Generic(R) : AlgMat -> AlgMat
BaseModule(R) : AlgMatV -> ModTup
NumberOfGenerators(R) : AlgMat -> { AlgMatElt }
Parent(a) : AlgMatElt -> AlgMat
Example AlgMat_Invariants (H92E3)
Construction of Subalgebras, Ideals and Quotient Rings
sub<R | L> : AlgMat, List -> AlgMat, Hom(Alg)
ideal<R | L> : AlgMat, List -> AlgMat
lideal<R | L> : AlgMat, List -> AlgMat
rideal<R | L> : AlgMat, List -> AlgMat
Example AlgMat_SubAlgebra (H92E4)
The Construction of Extensions and their Elements
The Construction of Direct Sums and Tensor Products
DirectSum(R, T) : AlgMat, AlgMat -> AlgMat
TensorProduct(A, B) : AlgMat, AlgMat -> AlgMat
Example AlgMat_Products (H92E5)
Construction of Direct Sums and Tensor Products of Elements
DirectSum(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
ExteriorSquare(a) : AlgMatElt -> AlgMatElt
ExteriorPower(a,r) : AlgMat, RngIntElt -> AlgMatElt
SymmetricSquare(a) : AlgMatElt -> AlgMatElt
SymmetricPower(a,r) : AlgMatElt, RngIntElt -> AlgMatElt
TensorProduct(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
Operations on Matrix Algebras
Centre(A) : AlgMat -> AlgMat
Centralizer(A, S) : AlgMat, AlgMat -> AlgMat
Changing Rings
ChangeRing(A, S) : AlgMatV, Rng -> AlgMat, Map
ChangeRing(A, S, f) : AlgMatV, Rng, Map -> AlgMat, Map
hom< A -> B | f > : AlgMat, AlgMat, Map -> Map
Elementary Operations on Elements
Arithmetic
a + b : AlgMatElt, AlgMatElt -> AlgMatElt
a + t : AlgMatElt, RngElt -> AlgMatElt
- a : AlgMatElt -> AlgMatElt
a - b : AlgMatElt, AlgMatElt -> AlgMatElt
a - t : AlgMatElt, RngElt -> AlgMatElt
a * b : AlgMatElt, AlgMatElt -> AlgMatElt
a * b : AlgMatElt, Mtrx -> Mtrx
a * b : Mtrx, AlgMatElt -> Mtrx
t * a : RngElt, AlgMatElt -> AlgMatElt
u * a : ModTupRngElt, AlgMatElt -> ModTupElt
a ^ n : AlgMatElt, RngIntElt -> AlgMatElt
NumberOfColumns(a) : AlgMatElt -> RngIntElt
NumberOfRows(a) : AlgMatElt -> RngIntElt
Predicates
Comparison
a eq b : AlgMatElt, AlgMatElt -> BoolElt
a ne b : AlgMatElt, AlgMatElt -> BoolElt
Properties of Elements
IsDiagonal(a) : AlgMatElt -> BoolElt
IsMinusOne(a) : AlgMatElt -> BoolElt
IsOne(a) : AlgMatElt -> BoolElt
IsScalar(a) : AlgMatElt -> BoolElt
IsSymmetric(a) : AlgMatElt -> BoolElt
IsUnit(a) : AlgMatElt -> BoolElt
IsZero(a) : AlgMatElt -> BoolElt
IsNilpotent(a) : AlgMatElt -> BoolElt, RngIntElt
IsUnipotent(a) : AlgMatElt -> BoolElt, RngIntElt
Rank(a) : AlgMatElt -> RngIntElt
Determinant(A) : AlgMatElt -> RngElt
Trace(a) : AlgMatElt -> RngElt
Transpose(a) : AlgMatElt -> AlgMatElt
Order(a) : AlgMatElt -> RngIntElt
FactoredOrder(a) : AlgMatElt -> [ <RngIntElt, RngIntElt> ]
ProjectiveOrder(a) : AlgMatElt -> RngIntElt
FactoredProjectiveOrder(a) : AlgMatElt -> [ <RngIntElt, RngIntElt> ]
CharacteristicPolynomial(a: parameters) : AlgMatElt -> RngUPolElt
MinimalPolynomial(a) : AlgMatElt -> RngUPolElt
HessenbergForm(a) : AlgMatElt -> AlgMatElt
Adjoint(a) : AlgMatElt -> AlgMatElt
Eigenvalues(a) : AlgMatElt -> { <FldElt, RngIntElt> }
Eigenspace(a, e) : AlgMatElt, FldElt -> ModTup
Elements of Mn as Homomorphisms
Image(a) : AlgMatElt -> ModTup
Kernel(a) : AlgMatElt -> ModTup
RowNullSpace(a) : AlgMatElt -> ModTup
Restrict(a, V) : AlgMatElt, ModTupRng -> AlgMatElt
Elementary Operations on Subalgebras and Ideals
Bases
Dimension(R) : AlgMatV -> RngIntElt
Basis(R) : AlgMatV -> [ AlgMatElt ]
BasisElement(R, i) : AlgMatV, RngIntElt -> AlgMatElt
Coordinates(R, X) : AlgMatV, AlgMatVElt -> [ RngElt ]
Intersection of Subalgebras
R meet T : AlgMat, AlgMat -> AlgMat
Membership and Equality
x in R : AlgMatElt, AlgMat -> BoolElt
x notin R : AlgMatElt, AlgMat -> BoolElt
R eq T : AlgMat, AlgMat -> BoolElt
R ne T : AlgMat, AlgMat -> BoolElt
Accessing and Modifying a Matrix
Indexing
a[i] : AlgMatElt, RngIntElt -> ModTupElt
a[i] := u : AlgMatElt, RngIntElt, RngElt -> AlgMatElt
a[i, j] : AlgMatElt, RngIntElt, RngIntElt -> RngElt
a[i, j] := t : AlgMatElt, RngIntElt, RngIntElt, RngElt -> AlgMatElt
ElementToSequence(a) : AlgMatElt -> [ RngElt ]
Extracting and Inserting Blocks
Submatrix(a, i, j, p, q) : Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx
InsertBlock(~a, b, i, j) : Mtrx, Mtrx, RngIntElt, RngIntElt -> Mtrx
Joining Matrices
HorizontalJoin(X, Y) : Mtrx, Mtrx -> Mtrx
HorizontalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt
VerticalJoin(X, Y) : ModMatRngElt, ModMatRngElt -> ModMatRngElt
VerticalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt
DiagonalJoin(X, Y) : ModMatRngElt, ModMatRngElt -> ModMatRngElt
DiagonalJoin(Q) : [ ModMatRngElt ] -> ModMatRngElt
Row and Column Operations
SwapRows(~a, i, j) : AlgMatElt, RngIntElt, RngIntElt ->
MultiplyRow(~a, u, j) : AlgMatElt, RngElt, RngIntElt ->
AddRow(~a, u, i, j) : AlgMatElt, RngElt, RngIntElt, RngIntElt ->
SwapColumns(~a, i, j) : AlgMatElt, RngIntElt, RngIntElt ->
MultiplyColumn(~a, u, i) : AlgMatElt, RngElt, RngIntElt ->
AddColumn(~a, u, i, j) : AlgMatElt, RngElt, RngIntElt, RngIntElt ->
Canonical Forms
Canonical Forms for Matrices over Euclidean Domains
EchelonForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt
ElementaryDivisors(a) : AlgMatElt -> [RngElt]
HermiteForm(X) : AlgMatElt -> AlgMatElt, AlgMatElt
SmithForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, AlgMatElt
Example AlgMat_EchelonForm (H92E6)
Canonical Forms for Matrices over a Field
PrimaryRationalForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]
JordanForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]
RationalForm(a) : AlgMatElt -> AlgMatElt, AlgMatElt, [ RngUPolElt ]
PrimaryInvariantFactors(a) : AlgMatElt -> [ <RngUPolElt, RngIntElt ]
InvariantFactors(a) : AlgMatElt -> [ AlgPolElt ]
IsSimilar(a, b) : AlgMatElt, AlgMatElt -> BoolElt, AlgMatElt
Example AlgMat_ElementaryDivisors (H92E7)
Example AlgMat_CanonicalForms (H92E8)
Diagonalising Commutative Algebras over a Field
CommonEigenspaces(Q) : [AlgMatElt] -> [**], [[FldElt]]
CommonEigenspaces(A) : AlgMat -> [**], [[FldElt]]
IsEtale(A) : AlgMat -> BoolElt, AlgMatElt
Diagonalisation(Q) : [AlgMatElt] -> [AlgMatElt], AlgMatElt
Diagonalisation(A) : AlgMat -> AlgMat, AlgMatElt
Diagonalisation(M) : AlgMatElt -> AlgMatElt, AlgMatElt
IsDiagonalisable(M) : AlgMatElt -> Boolelt, AlgMatElt, AlgMatElt
Example AlgMat_Diagonalization (H92E9)
Example AlgMat_IsDiagonalizablex (H92E10)
Solutions of Systems of Linear Equations
IsConsistent(A, w) : ModMatRngElt, ModTupRng -> BoolElt, ModTupRngElt, ModTupRng
IsConsistent(A, W) : ModMatRngElt, [ ModTupRng ] -> BoolElt, [ ModTupRngElt ], ModTupRng
Solution(A, w) : ModMatRngElt, ModTupRng -> ModTupRngElt, ModTupRng
Solution(A, W) : ModMatRngElt, [ ModTupRng ] -> [ ModTupRngElt ], ModTupRng
Presentations for Matrix Algebras
Quotients and Idempotents
NaturalFreeAlgebraCover(A) : AlgMat -> Map
SimpleQuotientAlgebras(A) : AlgMat -> Rec
PrimitiveIdempotentData(A) : AlgMat -> SeqEnum, Map, SeqEnum
PrimitiveIdempotents(A) : AlgMat -> SeqEnum
RanksOfPrimitiveIdempotents(A) : AlgMat -> SeqEnum
NaturalFreeAlgebraCover(A) : AlgMat -> Map
CondensedAlgebra(A) : AlgMat -> AlgMat
Example AlgMat_PrimitiveIdempotents (H92E11)
Generators and Presentations
SemisimpleGeneratorData(A) : AlgMat -> SeqEnum
AlgebraGenerators(A) : AlgMat -> Rec
AlgebraStructure(A) : AlgMat -> Rec
Presentation(A) : AlgMat -> AlgFr, AlgFr, Map
StandardFormConjugationMatrices(A) : AlgMat -> Tup
CondensationMatrices(A) : AlgMat -> Tup
SequenceOfRadicalGenerators(A) : AlgMat -> SeqEnum
CartanMatrix(A) : AlgMat -> ModMatRngElt
Example AlgMat_CondensedAlgebra (H92E12)
Solving the Word Problem
WordProblemData(A) : AlgMat -> List
WordProblem(A, x) : AlgMat, AlgMatElt -> BoolElt, AlgFrElt
Example AlgMat_Presentation (H92E13)
Bibliography
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