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MATRICES

 
Acknowledgements
 
Introduction
 
Creation of Matrices
      General Matrix Construction
      Shortcuts
      Construction of Structured Matrices
      Construction of Random Matrices
      Creating Vectors
 
Elementary Properties
 
Accessing or Modifying Entries
      Indexing
      Extracting and Inserting Blocks
      Row and Column Operations
 
Building Block Matrices
 
Changing Ring
 
Elementary Arithmetic
 
Nullspaces and Solutions of Systems
 
Predicates
 
Determinant and Other Properties
 
Minimal and Characteristic Polynomials and Eigenvalues
 
Canonical Forms
      Canonical Forms over General Rings
      Canonical Forms over Fields
      Canonical Forms over Euclidean Domains
 
Orders of Invertible Matrices
 
Miscellaneous Operations on Matrices
 
Bibliography







DETAILS

 
Introduction

 
Creation of Matrices

      General Matrix Construction
            Matrix(R, m, n, Q) : Rng, RngIntElt, RngIntElt, [ RngElt ] -> Mtrx
            Example Mat_Create (H26E1)

      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 (H26E2)

      Construction of Structured Matrices
            ZeroMatrix(R, m, n) : Rng, RngIntElt, 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 (H26E3)

      Construction of Random Matrices
            RandomMatrix(R, m, n) : Rng, RngIntElt, RngIntElt -> Mtrx
            RandomUnimodularMatrix(M, n) : 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

      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 (H26E4)

      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 (H26E5)

      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 (H26E6)

 
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


      DiagonalJoin(X, Y) : Mtrx, Mtrx -> Mtrx
      DiagonalJoin(Q) : [ Mtrx ] -> Mtrx
      KroneckerProduct(A, B) : Mtrx, Mtrx -> Mtrx

 
Changing Ring
      ChangeRing(A, R) : Mtrx, Rng -> Mtrx
      ChangeRing(A, R, f) : Mtrx, Rng, Map -> 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 (H26E7)
      Example Mat_Solution (H26E8)

 
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, RngIntElt -> 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 (H26E9)

      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 (H26E10)

 
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

 
Miscellaneous Operations on Matrices
      FrobeniusImage(A, e) : Mtrx, RngIntElt -> Mtrx
      NumericalEigenvectors(M, e) : Mtrx, FldComElt -> SeqEnum

 
Bibliography

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