Matrix Algebras
- Introduction
- Construction of Matrix Algebras and their Elements
- Construction of the Complete Matrix Algebra
- Construction of a Matrix
elt< R | L >: AlgMat, RngElt → AlgMatEltR ! Q: AlgMat, [ RngElt ] → AlgMatEltCambridgeMatrix(t, K, n, Q): RngIntElt, FldFin, RngIntElt, [ ] → AlgMatEltCompanionMatrix(p): RngUPolElt → AlgMatEltDiagonalMatrix(R, Q): AlgMat, [ RngElt ] → AlgMatEltMatrixUnit(R, i, j): AlgMat, RngIntElt, RngIntElt → AlgMatEltRandom(R): AlgMat → AlgMatEltScalarMatrix(R, t): AlgMat, RngElt → AlgMatEltR ! 1: AlgMat, RngIntElt → AlgMatEltR ! 0: AlgMat, RngIntElt → AlgMatEltR ! t: AlgMat, RngIntElt → AlgMatElt
- 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
- Operations on Matrix Algebras
- Changing Rings
ChangeRing(A, S): AlgMatV, Rng → AlgMat, MapChangeRing(A, S, f): AlgMatV, Rng, Map → AlgMat, Maphom< A -> B | f >: AlgMat, AlgMat, Map → Map
- Elementary Operations on Elements
- Arithmetic
a + b: AlgMatElt, AlgMatElt → AlgMatElta + t: AlgMatElt, RngElt → AlgMatEltt + a: AlgMatElt, RngElt → AlgMatElt- a: AlgMatElt → AlgMatElta - b: AlgMatElt, AlgMatElt → AlgMatElta - t: AlgMatElt, RngElt → AlgMatEltt - a: AlgMatElt, RngElt → AlgMatElta * b: AlgMatElt, AlgMatElt → AlgMatElta * b: AlgMatElt, Mtrx → Mtrxa * b: Mtrx, AlgMatElt → Mtrxt * a: RngElt, AlgMatElt → AlgMatElta * t: AlgMatElt, RngElt → AlgMatEltu * a: ModTupRngElt, AlgMatElt → ModTupElta ^ n: AlgMatElt, RngIntElt → AlgMatEltNumberOfColumns(a): AlgMatElt → RngIntEltNcols(a): AlgMatElt → RngIntEltNumberOfRows(a): AlgMatElt → RngIntEltNrows(a): AlgMatElt → RngIntElt
- Predicates
- Elements of \(M_n\) as Homomorphisms
- Elementary Operations on Subalgebras and Ideals
- Accessing and Modifying a Matrix
- Indexing
a[i]: AlgMatElt, RngIntElt → ModTupElta[i] := u: AlgMatElt, RngIntElt, RngElt → AlgMatElta[i, j]: AlgMatElt, RngIntElt, RngIntElt → RngElta[i, j] := t: AlgMatElt, RngIntElt, RngIntElt, RngElt → AlgMatEltElementToSequence(a): AlgMatElt → [ RngElt ]Eltseq(a): AlgMatElt → [ RngElt ]
- Extracting and Inserting Blocks
Submatrix(a, i, j, p, q): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt → MtrxExtractBlock(a, i, j, p, q): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt → MtrxInsertBlock(~a, b, i, j): Mtrx, Mtrx, RngIntElt, RngIntElt → Mtrx
- Joining Matrices
- Row and Column Operations
SwapRows(~a, i, j): AlgMatElt, RngIntElt, RngIntEltMultiplyRow(~a, u, j): AlgMatElt, RngElt, RngIntEltAddRow(~a, u, i, j): AlgMatElt, RngElt, RngIntElt, RngIntEltSwapColumns(~a, i, j): AlgMatElt, RngIntElt, RngIntEltMultiplyColumn(~a, u, i): AlgMatElt, RngElt, RngIntEltAddColumn(~a, u, i, j): AlgMatElt, RngElt, RngIntElt, RngIntElt
- Canonical Forms
- Canonical Forms for Matrices over Euclidean Domains
- 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: Elementary Divisors
- Example: Canonical Forms over a Finite Field
- Diagonalising Commutative Algebras over a Field
CommonEigenspaces(Q): [AlgMatElt] → [**], [[FldElt]]CommonEigenspaces(A): AlgMat → [**], [[FldElt]]IsEtale(A): AlgMat → BoolElt, AlgMatEltDiagonalisation(Q): [AlgMatElt] → [AlgMatElt], AlgMatEltDiagonalization(Q): [AlgMatElt] → [AlgMatElt], AlgMatEltDiagonalisation(A): AlgMat → AlgMat, AlgMatEltDiagonalization(A): AlgMat → AlgMat, AlgMatEltDiagonalisation(M): AlgMatElt → AlgMatElt, AlgMatEltDiagonalization(M): AlgMatElt → AlgMatElt, AlgMatEltIsDiagonalisable(M): AlgMatElt → Boolelt, AlgMatElt, AlgMatEltIsDiagonalizable(M): AlgMatElt → Boolelt, AlgMatElt, AlgMatElt- Example: Common Eigenspaces of Commuting Matrices
- Example: Diagonalising over an Extension Field
- Solutions of Systems of Linear Equations
IsConsistent(A, w): ModMatRngElt, ModTupRng → BoolElt, ModTupRngElt, ModTupRngIsConsistent(A, W): ModMatRngElt, [ ModTupRng ] → BoolElt, [ ModTupRngElt ], ModTupRngSolution(A, w): ModMatRngElt, ModTupRng → ModTupRngElt, ModTupRngSolution(A, W): ModMatRngElt, [ ModTupRng ] → [ ModTupRngElt ], ModTupRng
- Presentations for Matrix Algebras
- Bibliography