MATRIX ALGEBRAS

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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 M_n 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

DETAILS

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 (H83E1)
Example AlgMat_Cambridge (H83E2)

The Invariants of a Matrix Algebra
R . i : AlgMat, RngIntElt -> AlgMatElt
BaseRing(R) : AlgMat -> Rng
Degree(R) : AlgMat -> RngIntElt
Generators(R) : AlgMat -> { AlgMatElt }
Generic(R) : AlgMat -> AlgMat
BaseModule(R) : AlgMat -> ModTup
NumberOfGenerators(R) : AlgMat -> { AlgMatElt }
Parent(a) : AlgMatElt -> AlgMat
Example AlgMat_Invariants (H83E3)

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 (H83E4)

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 (H83E5)

Construction of Direct Sums and Tensor Products of Elements
DirectSum(a, b) : AlgMatElt, AlgMatElt -> AlgMatElt
ExteriorSquare(a) : AlgMat -> 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) : AlgMat, Rng -> AlgMat, Map
ChangeRing(A, S, f) : AlgMat, 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

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 M_n as Homomorphisms
Image(a) : AlgMatElt -> ModTup
Kernel(a) : AlgMatElt -> ModTup
RowNullSpace(a) : AlgMatElt -> ModTup

Elementary Operations on Subalgebras and Ideals

Bases
Dimension(R) : AlgMat -> RngIntElt
Basis(R) : AlgMat -> [ AlgMatElt ]
BasisElement(R, i) : AlgMat, RngIntElt -> AlgMatElt
Coordinates(R, X) : AlgMat, AlgMatElt -> [ 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 (H83E6)

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 (H83E7)
Example AlgMat_CanonicalForms (H83E8)

Diagonalising Commutative Algebras over a Field
CommonEigenspaces(Q) : [AlgMatElt] -> [**], [[FldElt]]
CommonEigenspaces(A) : AlgMat -> [**], [[FldElt]]
Diagonalisation(Q) : [AlgMatElt] -> [AlgMatElt], AlgMatElt
Diagonalisation(A) : AlgMat -> AlgMat, AlgMatElt
Example AlgMat_Diagonalization (H83E9)

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 (H83E10)

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 (H83E11)

Solving the Word Problem
WordProblemData(A) : AlgMat -> List
WordProblem(A, x) : AlgMat -> BoolElt, AlgFrElt
Example AlgMat_Presentation (H83E12)

Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013