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 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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Mon Dec 17 14:40:36 EST 2012