Matrices
- Introduction
- Creation of Matrices
- General Matrix Construction
Matrix(R, m, n, Q): Rng, RngIntElt, RngIntElt, [ RngElt ] → MtrxMatrix(R, m, n, Q): Rng, RngIntElt, RngIntElt, [ <RngIntElt, RngIntElt, RngElt> ] → MtrxMatrix(R, m, n, Q): Rng, RngIntElt, RngIntElt, [ [ RngElt ] ] → MtrxExample: Create
- Shortcuts
Matrix(m, n, Q): RngIntElt, RngIntElt, [ RngElt ] → MtrxMatrix(m, n, Q): RngIntElt, RngIntElt, [ [ RngElt ] ] → MtrxMatrix(Q): [ Mtrx ] → MtrxMatrix(R, n, Q): Rng, RngIntElt, [ RngElt ] → MtrxMatrix(n, Q): RngIntElt, [ RngElt ] → MtrxMatrix(Q): [ [ RngElt ] ] → MtrxMatrix(R, Q): Rng, [ [ RngElt ] ] → MtrxExample: Short Cuts
- Construction of Structured Matrices
ZeroMatrix(R, m, n): Rng, RngIntElt, RngIntElt → MtrxIdentityMatrix(R, n): Rng, RngIntElt → MtrxScalarMatrix(n, s): RngIntElt, RngElt → MtrxScalarMatrix(R, n, s): Rng, RngIntElt, RngElt → MtrxDiagonalMatrix(R, n, Q): Rng, RngIntElt, [ RngElt ] → MtrxDiagonalMatrix(R, Q): Rng, [ RngElt ] → MtrxDiagonalMatrix(Q): [ RngElt ] → MtrxMatrix(A): Mtrx → MtrxLowerTriangularMatrix(Q): [ RngElt ] → MtrxLowerTriangularMatrix(R, Q): Rng, [ RngElt ] → MtrxUpperTriangularMatrix(Q): [ RngElt ] → MtrxUpperTriangularMatrix(R, Q): Rng, [ RngElt ] → MtrxSymmetricMatrix(Q): [ RngElt ] → MtrxSymmetricMatrix(R, Q): Rng, [ RngElt ] → MtrxAntisymmetricMatrix(Q): [ RngElt ] → MtrxAntisymmetricMatrix(R, Q): Rng, [ RngElt ] → MtrxPermutationMatrix(R, Q): Rng, [ RngIntElt ] → MtrxPermutationMatrix(R, x): Rng, GrpPermElt → MtrxExample: Shortcuts
- Construction of Random Matrices
RandomMatrix(R, m, n): Rng, RngIntElt, RngIntElt → MtrxRandomUnimodularMatrix(n, M): RngIntElt, RngIntElt → MtrxRandomSLnZ(n, k, l): RngIntElt, RngIntElt, RngIntElt → AlgMatEltRandomGLnZ(n, k, l): RngIntElt, RngIntElt, RngIntElt → AlgMatEltRandomSymplecticMatrix(g, m): RngIntElt, RngIntElt → MtrxRandomSymmetricMatrix(R, n): Rng, RngIntElt → AlgMatEltRandomSymmetricMatrix(R, n, M): RngOrd, RngIntElt, RngIntElt → AlgMatEltRandomPositiveDefiniteSymmetricMatrix(n, M): RngIntElt, RngIntElt → AlgMatElt
- Creating Vectors
- Elementary Properties
- Accessing or Modifying Entries
- Indexing
A[i]: Mtrx, RngIntElt → ModTupRngEltA[i, j]: Mtrx, RngIntElt, RngIntElt → RngEltA[Q]: Mtrx, [ RngIntElt ] → RngEltA[i .. j]: Mtrx, RngIntElt, RngIntElt → RngEltA[i] := v: Mtrx, RngIntElt, MtrxA[i, j] := x: Mtrx, RngIntElt, RngIntElt, RngEltExample: Indexing
- 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 → MtrxSubmatrixRange(A, i, j, r, s): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt → MtrxExtractBlockRange(A, i, j, r, s): Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt → MtrxSubmatrix(A, I, J): Mtrx, [RngIntElt], [RngIntElt] → MtrxInsertBlock(A, B, i, j): Mtrx, Mtrx, RngIntElt, RngIntElt → MtrxInsertBlock(~A, B, i, j): Mtrx, Mtrx, RngIntElt, RngIntElt → MtrxRowSubmatrix(A, i, k): Mtrx, RngIntElt, RngIntElt → MtrxRowSubmatrix(A, i): Mtrx, RngIntElt → MtrxRowSubmatrixRange(A, i, j): Mtrx, RngIntElt, RngIntElt → MtrxColumnSubmatrix(A, i, k): Mtrx, RngIntElt, RngIntElt → MtrxColumnSubmatrix(A, i): Mtrx, RngIntElt → MtrxColumnSubmatrixRange(A, i, j): Mtrx, RngIntElt, RngIntElt → MtrxExample: Submatrix
- Row and Column Operations
SwapRows(A, i, j): Mtrx, RngIntElt, RngIntElt → MtrxSwapRows(~A, i, j): Mtrx, RngIntElt, RngIntEltSwapColumns(A, i, j): Mtrx, RngIntElt, RngIntElt → MtrxSwapColumns(~A, i, j): Mtrx, RngIntElt, RngIntEltReverseRows(A): Mtrx → MtrxReverseRows(~A): MtrxReverseColumns(A): Mtrx → MtrxReverseColumns(~A): MtrxAddRow(A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt → MtrxAddRow(~A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntEltAddColumn(A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntElt → MtrxAddColumn(~A, c, i, j): Mtrx, RngElt, RngIntElt, RngIntEltMultiplyRow(A, c, i): Mtrx, RngElt, RngIntElt → MtrxMultiplyRow(~A, c, i): Mtrx, RngElt, RngIntEltMultiplyColumn(A, c, i): Mtrx, RngElt, RngIntElt → MtrxMultiplyColumn(~A, c, i): Mtrx, RngElt, RngIntEltRemoveRow(A, i): Mtrx, RngIntElt → MtrxRemoveRow(~A, i): Mtrx, RngIntEltRemoveColumn(A, j): Mtrx, RngIntElt → MtrxRemoveColumn(~A, j): Mtrx, RngIntEltRemoveRowColumn(A, i, j): Mtrx, RngIntElt → MtrxRemoveRowColumn(~A, i, j): Mtrx, RngIntEltRemoveZeroRows(A): Mtrx → MtrxRemoveZeroRows(~A): MtrxExample: Row Column Ops
- Building Block Matrices
BlockMatrix(m, n, blocks): RngIntElt, RngIntElt, [ Mtrx ] → MtrxBlockMatrix(m, n, rows): RngIntElt, RngIntElt, [ [ Mtrx ] ] → MtrxBlockMatrix(rows): [ [ Mtrx ] ] → MtrxHorizontalJoin(X, Y): Mtrx, Mtrx → MtrxHorizontalJoin(Q): [ Mtrx ] → MtrxHorizontalJoin(T): < Mtrx > → MtrxVerticalJoin(X, Y): Mtrx, Mtrx → MtrxVerticalJoin(Q): [ Mtrx ] → MtrxVerticalJoin(T): < Mtrx > → MtrxDiagonalJoin(X, Y): Mtrx, Mtrx → MtrxDiagonalJoin(Q): [ Mtrx ] → MtrxDiagonalJoin(T): < Mtrx > → MtrxKroneckerProduct(A, B): Mtrx, Mtrx → Mtrx
- Changing Ring
ChangeRing(A, R): Mtrx, Rng → MtrxMatrix(R, A): Rng, Mtrx → MtrxChangeRing(A, R, f): Mtrx, Rng, Map → MtrxChangeRing(A, f): Mtrx, Map → MtrxCanChangeRing(A, R): Mtrx, Rng → BoolElt, Mtrx
- Elementary Arithmetic
A + B: Mtrx, Mtrx → MtrxA - B: Mtrx, Mtrx → MtrxA * B: Mtrx, Mtrx → Mtrxx * A: RngElt, Mtrx → MtrxA * x: Mtrx, RngElt → Mtrx- A: Mtrx → MtrxA ^ -1: Mtrx, RngIntElt → MtrxA ^ n: Mtrx, RngIntElt → MtrxTranspose(A): Mtrx → MtrxAddScaledMatrix(A, s, B): Mtrx, RngElt, Mtrx → MtrxAddScaledMatrix(~A, s, B): Mtrx, RngElt, Mtrx
- Nullspaces and Solutions of Systems
Nullspace(A): Mtrx → ModTupRngKernel(A): Mtrx → ModTupRng, MapNullspaceMatrix(A): Mtrx → ModTupRngKernelMatrix(A): Mtrx → ModTupRngNullspaceOfTranspose(A): Mtrx → ModTupRngIsConsistent(A, W): Mtrx, Mtrx → BoolElt, Mtrx, ModTupRngIsConsistent(A, Q): Mtrx, [ ModTupRng ] → BoolElt, [ ModTupRngElt ], ModTupRngSolution(A, W): ModMatRngElt, ModTupRng → ModTupRngElt, ModTupRngSolution(A, Q): ModMatRngElt, [ ModTupRng ] → [ ModTupRngElt ], ModTupRngExample: NullspaceExample: Solution
- Predicates
- Determinant and Other Properties
Determinant(A: parameters): Mtrx → RngEltTrace(A): Mtrx → RngEltTraceOfProduct(A, B): Mtrx, Mtrx → RngEltRank(A): Mtrx → RngIntEltMinor(M, i, j): Mtrx, RngIntElt, RngIntElt → RngEltMinor(M, I, J): Mtrx, [RngIntElt], [RngIntElt] → RngEltMinors(M, r): Mtrx, RngIntElt → SeqEnumCofactor(M, i, j): Mtrx, RngIntElt, RngIntElt → RngEltCofactors(M): Mtrx → SeqEnumCofactors(M, r): Mtrx, RngIntElt → SeqEnumPfaffian(M): Mtrx → RngEltPfaffian(M, I, J): Mtrx, [RngIntElt], [RngIntElt] → RngEltPfaffians(M, r): Mtrx, RngIntElt → SeqEnum
- Minimal and Characteristic Polynomials and Eigenvalues
MinimalPolynomial(A: parameters): Mtrx → RngUPolEltCharacteristicPolynomial(A: parameters): Mtrx → RngUPolEltMinimalAndCharacteristicPolynomials(A: parameters): Mtrx → RngUPolElt, RngUPolEltMCPolynomials(A): Mtrx → RngUPolElt, RngUPolEltFactoredMinimalPolynomial(A: parameters): Mtrx → [ <RngUPolElt, RngIntElt>]FactoredCharacteristicPolynomial(A: parameters): Mtrx → [ <RngUPolElt, RngIntElt>]FactoredMinimalAndCharacteristicPolynomials(A: parameters): Mtrx → [<RngUPolElt, RngIntElt>], [<RngUPolElt, RngIntElt>]FactoredMCPolynomials(A: parameters): Mtrx → [<RngUPolElt, RngIntElt>], [<RngUPolElt, RngIntElt>]Eigenvalues(A): Mtrx → { <FldElt, RngIntElt> }Eigenspace(A, e): AlgMatElt, FldElt → ModTup
- Canonical Forms
- Canonical Forms over General Rings
- Canonical Forms over Fields
PrimaryRationalForm(A): Mtrx → AlgMatElt, AlgMatElt, [ <RngUPolElt, RngIntElt ]JordanForm(A): Mtrx → Mtrx, AlgMatElt, [ <RngUPolElt, RngIntElt> ]RationalForm(A): Mtrx → Mtrx, AlgMatElt, [ RngUPolElt ]PrimaryInvariantFactors(A): Mtrx → [ <RngUPolElt, RngIntElt> ]InvariantFactors(A): Mtrx → [ RngUPolElt ]IsSimilar(A, B): AlgMatElt, AlgMatElt → BoolElt, AlgMatEltHessenbergForm(A): Mtrx → AlgMatEltFrobeniusFormAlternating(A): AlgMatElt → SeqEnumExample: Canonical Forms
- Canonical Forms over Euclidean Domains
- Orders of Invertible Matrices
- Numerical Linear Algebra
- Miscellaneous Operations on Matrices
