Introduction

 
Creation of Matrices

      General Matrix Construction
            Matrix(R, m, n, Q) : Rng, RngIntElt, RngIntElt, [ RngElt ] -> Mtrx
            Example Mat_Create (H27E1)

      Shortcuts
            Matrix(m, n, Q) : RngIntElt, RngIntElt, [ RngElt ] -> Mtrx
            Matrix(m, n, Q) : RngIntElt, RngIntElt, [ [ RngElt ] ] -> Mtrx
            Matrix(Q) : [ Mtrx ] -> Mtrx
            Matrix(R, n, Q) : Rng, RngIntElt, [ RngElt ] -> Mtrx
            Matrix(n, Q) : RngIntElt, [ RngElt ] -> Mtrx
            Matrix(Q) : [ [ RngElt ] ] -> Mtrx
            Matrix(R, Q) : Rng, [ [ RngElt ] ] -> Mtrx
            Example Mat_ShortCuts (H27E2)

      Construction of Structured Matrices
            ZeroMatrix(R, m, n) : Rng, RngIntElt, RngIntElt -> Mtrx
            IdentityMatrix(R, n) : Rng, RngIntElt -> Mtrx
            ScalarMatrix(n, s) : RngIntElt, RngElt -> Mtrx
            ScalarMatrix(R, n, s) : Rng, RngIntElt, RngElt -> Mtrx
            DiagonalMatrix(R, n, Q) : Rng, RngIntElt, [ RngElt ] -> Mtrx
            DiagonalMatrix(R, Q) : Rng, [ RngElt ] -> Mtrx
            DiagonalMatrix(Q) : [ RngElt ] -> Mtrx
            Matrix(A) : Mtrx -> Mtrx
            LowerTriangularMatrix(Q) : [ RngElt ] -> Mtrx
            LowerTriangularMatrix(R, Q) : Rng, [ RngElt ] -> Mtrx
            UpperTriangularMatrix(Q) : [ RngElt ] -> Mtrx
            UpperTriangularMatrix(R, Q) : Rng, [ RngElt ] -> Mtrx
            SymmetricMatrix(Q) : [ RngElt ] -> Mtrx
            SymmetricMatrix(R, Q) : Rng, [ RngElt ] -> Mtrx
            AntisymmetricMatrix(Q) : [ RngElt ] -> Mtrx
            AntisymmetricMatrix(R, Q) : Rng, [ RngElt ] -> Mtrx
            PermutationMatrix(R, Q) : Rng, [ RngIntElt ] -> Mtrx
            PermutationMatrix(R, x) : Rng, GrpPermElt -> Mtrx
            Example Mat_Shortcuts (H27E3)

      Construction of Random Matrices
            RandomMatrix(R, m, n) : Rng, RngIntElt, RngIntElt -> Mtrx
            RandomUnimodularMatrix(n, M) : RngIntElt, RngIntElt -> Mtrx
            RandomSLnZ(n, k, l) : RngIntElt, RngIntElt, RngIntElt -> AlgMatElt
            RandomGLnZ(n, k, l) : RngIntElt, RngIntElt, RngIntElt -> AlgMatElt
            RandomSymplecticMatrix(g, m) : RngIntElt, RngIntElt -> Mtrx
            RandomSymmetricMatrix(R, n) : Rng, RngIntElt -> AlgMatElt
            RandomPositiveDefiniteSymmetricMatrix(n, M) : RngIntElt, RngIntElt -> AlgMatElt

      Creating Vectors
            Vector(n, Q) : RngIntElt, [ RngElt ] -> ModTupRngElt
            Vector(Q) : [ RngElt ] -> ModTupRngElt
            Vector(R, n, Q) : Rng, RngIntElt, [ RngElt ] -> ModTupRngElt
            Vector(R, Q) : Rng, [ RngElt ] -> ModTupRngElt

 
Elementary Properties
      NumberOfRows(A) : Mtrx -> RngIntElt
      NumberOfColumns(A) : Mtrx -> RngIntElt
      NumberOfNonZeroEntries(A) : Mtrx -> RngIntElt
      Density(A) : Mtrx -> FldRe
      BaseRing(A) : Mtrx -> Rng
      ElementToSequence(A) : Mtrx -> [ RngElt ]
      RowSequence(A) : Mtrx -> [ [RngElt] ]

 
Accessing or Modifying Entries

      Indexing
            A[i] : Mtrx, RngIntElt -> ModTupRngElt
            A[i, j] : Mtrx, RngIntElt, RngIntElt -> RngElt
            A[Q] : Mtrx, [ RngIntElt ] -> RngElt
            A[i] := v : Mtrx, RngIntElt, Mtrx ->
            A[i, j] := x : Mtrx, RngIntElt, RngIntElt, RngElt ->
            Example Mat_Indexing (H27E4)

      Extracting and Inserting Blocks
            Submatrix(A, i, j, p, q) : Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx
            SubmatrixRange(A, i, j, r, s) : Mtrx, RngIntElt, RngIntElt, RngIntElt, RngIntElt -> Mtrx
            Submatrix(A, I, J) : Mtrx, [RngIntElt], [RngIntElt] -> Mtrx
            InsertBlock(A, B, i, j) : Mtrx, Mtrx, RngIntElt, RngIntElt -> Mtrx
            RowSubmatrix(A, i, k) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            RowSubmatrix(A, i) : Mtrx, RngIntElt -> Mtrx
            RowSubmatrixRange(A, i, j) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            ColumnSubmatrix(A, i, k) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            ColumnSubmatrix(A, i) : Mtrx, RngIntElt -> Mtrx
            ColumnSubmatrixRange(A, i, j) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            Example Mat_Submatrix (H27E5)

      Row and Column Operations
            SwapRows(A, i, j) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            SwapColumns(A, i, j) : Mtrx, RngIntElt, RngIntElt -> Mtrx
            ReverseRows(A) : Mtrx -> Mtrx
            ReverseColumns(A) : Mtrx -> Mtrx
            AddRow(A, c, i, j) : Mtrx, RngElt, RngIntElt, RngIntElt -> Mtrx
            AddColumn(A, c, i, j) : Mtrx, RngElt, RngIntElt, RngIntElt -> Mtrx
            MultiplyRow(A, c, i) : Mtrx, RngElt, RngIntElt -> Mtrx
            MultiplyColumn(A, c, i) : Mtrx, RngElt, RngIntElt -> Mtrx
            RemoveRow(A, i) : Mtrx, RngIntElt -> Mtrx
            RemoveColumn(A, j) : Mtrx, RngIntElt -> Mtrx
            RemoveRowColumn(A, i, j) : Mtrx, RngIntElt -> Mtrx
            RemoveZeroRows(A) : Mtrx -> Mtrx
            Example Mat_RowColumnOps (H27E6)

 
Building Block Matrices
      BlockMatrix(m, n, blocks) : RngIntElt, RngIntElt, [ Mtrx ] -> Mtrx
      BlockMatrix(m, n, rows) : RngIntElt, RngIntElt, [ [ Mtrx ] ] -> Mtrx
      HorizontalJoin(X, Y) : Mtrx, Mtrx -> Mtrx
      HorizontalJoin(Q) : [ Mtrx ] -> Mtrx
      VerticalJoin(X, Y) : Mtrx, Mtrx -> Mtrx
      VerticalJoin(Q) : [ Mtrx ] -> Mtrx

 
Changing Ring
      ChangeRing(A, R) : Mtrx, Rng -> Mtrx
      ChangeRing(A, R, f) : Mtrx, Rng, Map -> Mtrx
      CanChangeRing(A, R) : Mtrx, Rng -> BoolElt, Mtrx

 
Elementary Arithmetic
      A + B : Mtrx, Mtrx -> Mtrx
      A - B : Mtrx, Mtrx -> Mtrx
      A * B : Mtrx, Mtrx -> Mtrx
      x * A : RngElt, Mtrx -> Mtrx
      - A : Mtrx -> Mtrx
      A ^ -1 : Mtrx, RngIntElt -> Mtrx
      A ^ n : Mtrx, RngIntElt -> Mtrx
      Transpose(A) : Mtrx -> Mtrx
      AddScaledMatrix(A, s, B) : Mtrx, RngElt, Mtrx -> Mtrx
      AddScaledMatrix(~A, s, B) : Mtrx, RngElt, Mtrx ->

 
Nullspaces and Solutions of Systems
      Nullspace(A) : Mtrx -> ModTupRng
      NullspaceMatrix(A) : Mtrx -> ModTupRng
      NullspaceOfTranspose(A) : Mtrx -> ModTupRng
      IsConsistent(A, W) : Mtrx, Mtrx -> BoolElt, Mtrx, ModTupRng
      IsConsistent(A, Q) : Mtrx, [ ModTupRng ] -> BoolElt, [ ModTupRngElt ], ModTupRng
      Solution(A, W) : ModMatRngElt, ModTupRng -> ModTupRngElt, ModTupRng
      Solution(A, Q) : ModMatRngElt, [ ModTupRng ] -> [ ModTupRngElt ], ModTupRng
      Example Mat_Nullspace (H27E7)
      Example Mat_Solution (H27E8)

 
Predicates
      IsZero(A) : Mtrx -> BoolElt
      IsOne(A) : Mtrx -> BoolElt
      IsMinusOne(A) : Mtrx -> BoolElt
      IsScalar(A) : Mtrx -> BoolElt
      IsDiagonal(A) : Mtrx -> BoolElt
      IsSymmetric(A) : Mtrx -> BoolElt
      IsUpperTriangular(A) : Mtrx -> BoolElt
      IsLowerTriangular(A) : Mtrx -> BoolElt
      IsUnit(A) : Mtrx -> BoolElt
      IsSingular(A) : Mtrx -> BoolElt
      IsSymplecticMatrix(A) : Mtrx -> BoolElt

 
Determinant and Other Properties
      Determinant(A: parameters) : Mtrx -> RngElt
      Trace(A) : Mtrx -> RngElt
      TraceOfProduct(A, B) : Mtrx, Mtrx -> RngElt
      Rank(A) : Mtrx -> RngIntElt
      Minor(M, i, j) : Mtrx, RngIntElt, RngIntElt -> RngElt
      Minor(M, I, J) : Mtrx, [RngIntElt], [RngIntElt] -> RngElt
      Minors(M, r) : Mtrx, RngIntElt -> SeqEnum
      Cofactor(M, i, j) : Mtrx, RngIntElt, RngIntElt -> RngElt
      Cofactors(M) : Mtrx -> SeqEnum
      Cofactors(M, r) : Mtrx, RngIntElt -> SeqEnum
      Pfaffian(M) : Mtrx -> RngElt

 
Minimal and Characteristic Polynomials and Eigenvalues
      MinimalPolynomial(A: parameters) : Mtrx -> RngUPolElt
      CharacteristicPolynomial(A: parameters) : Mtrx -> RngUPolElt
      MinimalAndCharacteristicPolynomials(A: parameters) : Mtrx -> RngUPolElt, RngUPolElt
      FactoredMinimalPolynomial(A: parameters) : Mtrx -> [ <RngUPolElt, RngIntElt>]
      FactoredCharacteristicPolynomial(A: parameters) : Mtrx -> [ <RngUPolElt, RngIntElt>]
      FactoredMinimalAndCharacteristicPolynomials(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
            EchelonForm(A) : Mtrx -> Mtrx, AlgMatElt
            Adjoint(A) : Mtrx -> AlgMatElt

      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, AlgMatElt
            HessenbergForm(A) : Mtrx -> AlgMatElt
            FrobeniusFormAlternating(A) : AlgMatElt -> SeqEnum
            Example Mat_CanonicalForms (H27E9)

      Canonical Forms over Euclidean Domains
            HermiteForm(A) : Mtrx -> Mtrx, ModMatRngElt
            SmithForm(A) : ModMatRngElt -> ModMatRngElt, ModMatRngElt, ModMatRngElt
            ElementaryDivisors(A) : Mtrx -> [RngElt]
            Saturation(A) : Mtrx -> Mtrx
            Example Mat_Forms1 (H27E10)

 
Orders of Invertible Matrices
      HasFiniteOrder(A) : Mtrx -> BoolElt
      Order(A) : AlgMatElt -> RngIntElt
      FactoredOrder(A) : AlgMatElt -> [ <RngIntElt, RngIntElt> ]
      ProjectiveOrder(A) : AlgMatElt -> RngIntElt, RngElt
      FactoredProjectiveOrder(A) : AlgMatElt -> [ <RngIntElt, RngIntElt> ], RngElt

 
Numerical Linear Algebra
      RQDecomposition(M) : Mtrx[RngReCom] -> Mtrx, AlgMatElt
      QLDecomposition(M) : Mtrx[RngReCom] -> AlgMatElt, Mtrx
      NumericalInverse(M) : Mtrx[RngReCom] -> AlgMatElt
      Example Mat_numerlinalg-examples (H27E11)

      Rank, Kernel, Solution, and Pseudoinverse
            NumericalRank(M) : Mtrx[RngReCom] -> RngIntElt
            NumericalKernel(M) : Mtrx[RngReCom] -> Mtrx
            NumericalImage(M) : Mtrx[RngReCom] -> Mtrx
            NumericalSolution(M,w) : Mtrx[RngReCom], Mtrx[RngReCom] -> Mtrx, Mtrx
            NumericalIsConsistent(M,w) : Mtrx[RngReCom], Mtrx[RngReCom] -> BoolElt, Mtrx, Mtrx
            NumericalPseudoinverse(M) : Mtrx[RngReCom] -> Mtrx
            Example Mat_numerlinalg-examples2 (H27E12)

      Eigenvalues and the Singular Value Decomposition
            NumericalHessenbergForm(M) : Mtrx[RngReCom] -> Mtrx, Mtrx
            NumericalSchurForm(M) : Mtrx[RngReCom] -> Mtrx, Mtrx
            NumericalEigenvalues(M) : Mtrx[RngReCom] -> SeqEnum
            NumericalEigenvectors(M, e) : Mtrx, FldComElt -> SeqEnum
            NumericalBidiagonalForm(M) : Mtrx[RngReCom] -> Mtrx, Mtrx, Mtrx
            NumericalSingularValueDecomposition(M) : Mtrx[RngReCom] -> Mtrx, Mtrx, Mtrx
            Example Mat_numerlinalg-examples3 (H27E13)
            Example Mat_numerlinalg-examples4 (H27E14)

 
Miscellaneous Operations on Matrices
      FrobeniusImage(A, e) : Mtrx, RngIntElt -> Mtrx

 
Bibliography

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