Multilinear Algebra
- Introduction
- Tensors
- Creating Tensors
- Black-box Tensors
Tensor(S, F): SeqEnum, UserProgram → TenSpcElt, ListTensor(S, F): List, UserProgram → TenSpcElt, ListTensor(S, F, Cat): SeqEnum, UserProgram, TenCat → TenSpcElt, ListTensor(S, F, Cat): List, UserProgram, TenCat → TenSpcElt, ListExample: BB Tensors FrameExample: BB Cross ProductTensor(D, C, F): SeqEnum, Any, UserProgram → TenSpcElt, ListTensor(D, C, F): List, Any, UserProgram → TenSpcElt, ListTensor(D, C, F, Cat): SeqEnum, Any, UserProgram, TenCat → TenSpcElt, ListTensor(D, C, F, Cat): List, Any, UserProgram, TenCat → TenSpcElt, ListExample: BB Triple Product
- Tensors with Structure Constant Sequences
Tensor(D, S): [RngIntElt], SeqEnum → TenSpcEltTensor(R, D, S): Rng, [RngIntElt], SeqEnum → TenSpcEltTensor(D, S, Cat): [RngIntElt], SeqEnum, TenCat → TenSpcEltTensor(R, D, S, Cat): Rng, [RngIntElt], SeqEnum, TenCat → TenSpcEltExample: SC TensorsStructureConstants(T): TenSpcElt → SeqEnumEltseq(T): TenSpcElt → SeqEnumAssign(T, ind, k): TenSpcElt, [RngIntElt], Any → TenSpcEltAssign(~T, ind, k): TenSpcElt, [RngIntElt], AnyExample: SC From BB TensorsExample: SC Stored
- Bilinear Tensors
Tensor(M, s, t): [Mtrx], RngIntElt, RngIntElt → TenSpcEltTensor(M, s, t, C): [Mtrx], RngIntElt, RngIntElt, TenCat → TenSpcEltExample: Symplectic FormAsMatrices(T, s, t): TenSpcElt, RngIntElt, RngIntElt → SeqEnumSystemOfForms(T): TenSpcElt → SeqEnumExample: Trilinear As Mats
- Tensors from Algebraic Objects
Tensor(A): Alg → TenSpcElt, MapExample: D4Lie AlgebraTensor(Q): RngUPolRes → TenSpcElt, MapExample: Witt AlgebraCommutatorTensor(A): Alg → TenSpcElt, MapAnticommutatorTensor(A): Alg → TenSpcElt, MapExample: Commutator From AlgebraExample: Matrix Jordan AlgebraAssociatorTensor(A): Alg → TenSpcElt, MapExample: Associator From AlgebrapCentralTensor(G, p, s, t): Grp, RngIntElt, RngIntElt, RngIntElt → TenSpcElt, ListpCentralTensor(G, s, t): GrpPC, RngIntElt, RngIntElt → TenSpcElt, ListpCentralTensor(G, s, t): Grp, RngIntElt, RngIntElt → TenSpcElt, ListpCentralTensor(G): Grp → TenSpcElt, ListExample: Tensor P GroupMatrixTensor(K, S): Fld, [RngIntElt] → TenSpcElt, ListPolarisation(f): MPolElt → TenSpcElt, MPolEltPolarisation(f): RngUPolElt → TenSpcEltPolarization(f): MPolElt → TenSpcElt, MPolEltPolarization(f): RngUPolElt → TenSpcEltExample: Tensor Polarization
- New Tensors from Old
- Operations with Tensors
- Elementary Operations
- General Properties
- Tensors As Multilinear Maps
- Operations with Bilinear Maps
x * T: Any, TenSpcElt → AnyT * y: TenSpcElt, Any → Anyx * T: Any, TenSpc → AnyT * y: TenSpc, Any → AnyExample: Bimap Infixx * y: BmpUElt, BmpVElt → Anyx * y: BmpU, BmpV → Anyx * y: BmpUElt, BmpV → Anyx * y: BmpU, BmpVElt → AnyLeftDomain(B): TenSpcElt → BmpURightDomain(B): TenSpcElt → BmpVIsCoercible(U,x): BmpU, Any → BoolElt, BmpUEltIsCoercible(V,x): BmpV, Any → BoolElt, BmpVEltU ! x: BmpU, Any → BmpUEltV ! x: BmpV, Any → BmpVEltExample: Bimap ProductParent(x): BmpUElt → BmpUParent(x): BmpVElt → BmpVParent(X): BmpU → TenSpcEltParent(X): BmpV → TenSpcEltu1 eq u2: BmpUElt, BmpUElt → BoolEltv1 eq v2: BmpUElt, BmpUElt → BoolEltU1 eq U2: BmpU, BmpU → BoolEltV1 eq V2: BmpV, BmpV → BoolEltExample: Bimap Product2
- Manipulating Tensor Data
Slice(T, grid): TenSpcElt, [SetEnum] → SeqEnumInducedTensor(T, grid): TenSpcElt, [SetEnum] → TenSpcEltExample: Tensor SlicingSliceAsMatrices(T, grid, a, b): TenSpcElt, [SetEnum], RngIntElt, RngIntElt → SeqEnumExample: Slice As MatricesFoliation(T, i): TenSpcElt, RngIntElt → MtrxExample: Exfoliate FoliationAsTensorSpace(T, i): TenSpcElt, RngIntElt → TenSpc, MtrxAsCotensorSpace(T): TenSpcElt → TenSpc, MtrxExample: Tensors To SpacesAsTensor(S): TenSpc → TenSpcEltExample: Spaces To Tensors
- Invariants of Tensors
Induce(X, i): AlgMat, RngIntElt → Map, AlgMatInduce(X, i): AlgMatLie, RngIntElt → Map, AlgMatLieInduce(X, i): GrpMat, RngIntElt → Map, GrpMatExample: InducingDerivedFrom(~X, t, C, RC : parameters): Any, TenSpcElt, {RngIntElt}, {RngIntElt}- Standard Invariants
- Exporting Tensors
- Tensor Spaces
- Constructions of Tensor and Cotensor Spaces
- Universal Tensor Spaces
KTensorSpace(K, S): Fld, [RngIntElt] → TenSpcKTensorSpace(K, S, C): Fld, [RngIntElt], TenCat → TenSpcRTensorSpace(R, S): Rng, [RngIntElt] → TenSpcRTensorSpace(R, S, C): Rng, [RngIntElt], TenCat → TenSpcExample: Universal K Ten SpcTensorSpace(S): SeqEnum → TenSpc, ListTensorSpace(S): List → TenSpc, ListTensorSpace(S, C): SeqEnum, TenCat → TenSpc, ListTensorSpace(S, C): List, TenCat → TenSpc, ListExample: Universal Ten SpcTensorSpace(V, p, q): ModTupFld, RngIntElt, RngIntElt → TenSpcTensorSpace(K, d, p, q): Fld, RngIntElt, RngIntElt, RngIntElt → TenSpcExample: Signatured Ten Spc
- Universal Cotensor Spaces
KCotensorSpace(K, S): Fld, [RngIntElt] → TenSpcKCotensorSpace(K, S, C): Fld, [RngIntElt], TenCat → TenSpcCotensorSpace(S): SeqEnum → TenSpcCotensorSpace(S): List → TenSpc, ListCotensorSpace(S, C): SeqEnum, TenCat → TenSpcCotensorSpace(S, C): List, TenCat → TenSpc, ListExample: Universal Co Ten Spc
- Some Standard Constructions
- Operations on Tensor Spaces
- Membership and Comparison with Tensor Spaces
T in TS: TenSpcElt, TenSpc → BoolEltTS ! T: TenSpc, TenSpcElt → TenSpcEltTS ! S: TenSpc, SeqEnum → TenSpcEltT ! n: TenSpc, RngIntElt → TenSpcEltIsCoercible(TS, x): TenSpc, Any → BoolElt, TenSpcEltExample: CoercionS eq T: TenSpc, TenSpc → BoolEltS subset T: TenSpc, TenSpc → BoolEltIsCoercible(T, S): TenSpc, Any → BoolEltT ! S: TenSpc, TenSpc → TenSpcExample: Ten Spc Containment
- Tensor Spaces as Modules
Basis(T): TenSpc → SeqEnumGenerators(T): TenSpc → SeqEnumT . i: TenSpc, RngIntElt → TenSpcEltNumberOfGenerators(T): TenSpc → RngIntEltNgens(T): TenSpc → RngIntEltDimension(T): TenSpc → RngIntElt# T: TenSpc → RngIntEltExample: Basic ModuleRandom(T): TenSpc → TenSpcEltRandomTensor(R, S): Rng, [RngIntElt] → TenSpcEltRandomTensor(R, S, C): Rng, [RngIntElt], TenCat → TenSpcEltRandomCotensor(K, S): Fld, [RngIntElt] → TenSpcEltExample: Random TensorsRandomAlternatingTensor(R, d, n, c): Rng, RngIntElt, RngIntElt, RngIntElt → TenSpcEltRandomAlternatingTensor(R, S): Rng, [RngIntElt] → TenSpcEltRandomAntisymmetricTensor(R, d, n, c): Rng, RngIntElt, RngIntElt, RngIntElt → TenSpcEltRandomAntisymmetricTensor(R, S): Rng, [RngIntElt] → TenSpcEltRandomSymmetricTensor(R, d, n, c): Rng, RngIntElt, RngIntElt, RngIntElt → TenSpcEltRandomSymmetricTensor(R, S): Rng, [RngIntElt] → TenSpcEltExample: Random Sym Ten
- Properties of Tensor Spaces
- Tensor Categories
- Constructing Tensor Categories
TensorCategory(A, P): [RngIntElt], {SetEnum} → TenCatTensorCategory(A, P): Map, {SetEnum} → TenCatCotensorCategory(A, P): [RngIntElt], {SetEnum} → TenCatCotensorCategory(A, P): Map, {SetEnum} → TenCatExample: Basic Cat ConstHomotopismCategory(v : parameters): RngIntElt → TenCatCohomotopismCategory(v): RngIntElt → TenCatAdjointCategory(v, s, t): RngIntElt, RngIntElt, RngIntElt → TenCatLinearCategory(v, s, t): RngIntElt, RngIntElt, RngIntElt → TenCatExample: Ten Cat Special
- Operations on Tensor Categories
- Categorical Operations
- Categorical Operations on Tensors
Subtensor(T, S): TenSpcElt, List → TenSpcEltSubtensor(T, S): TenSpcElt, SeqEnum → TenSpcEltSubtensor(T, D, C): TenSpcElt, List, Any → TenSpcEltSubtensor(T, D, C): TenSpcElt, SeqEnum, Any → TenSpcEltIsSubtensor(T, S): TenSpcElt, TenSpcElt → BoolEltExample: SubtensorsLocalIdeal(T, S, I): TenSpcElt, List, {RngIntElt} → TenSpcEltLocalIdeal(T, S, I): TenSpcElt, SeqEnum, {RngIntElt} → TenSpcEltLocalIdeal(T, D, C, I): TenSpcElt, List, Any, {RngIntElt} → TenSpcEltLocalIdeal(T, D, C, I): TenSpcElt, SeqEnum, Any, {RngIntElt} → TenSpcEltLocalIdeal(T, S, I): TenSpcElt, TenSpcElt, {RngIntElt} → TenSpcEltIsLocalIdeal(T, S, I): TenSpcElt, TenSpcElt, {RngIntElt} → BoolEltExample: Local IdealsIdeal(T, S): TenSpcElt, List → TenSpcEltIdeal(T, S): TenSpcElt, SeqEnum → TenSpcEltIdeal(T, D, C): TenSpcElt, List, Any → TenSpcEltIdeal(T, D, C): TenSpcElt, SeqEnum, Any → TenSpcEltIdeal(T, S): TenSpcElt, TenSpcElt → TenSpcEltIsIdeal(T, S): TenSpcElt, TenSpcElt → BoolEltExample: IdealsLocalQuotient(T, S, I : parameters): TenSpcElt, TenSpcElt, {RngIntElt} → TenSpcElt, HmtpQuotient(T, S : parameters): TenSpcElt, TenSpcElt → TenSpcElt, HmtpT / S: TenSpcElt, TenSpcElt → TenSpcElt, HmtpExample: Quotients
- Categorical Operations on Tensor Spaces
SubConstructor(T, L): TenSpc, Any → TenSpc, Mapsub< T | L >: TenSpc, Any → TenSpc, MapIsSubtensorSpace(T, S): TenSpc, TenSpc → BoolEltExample: Subtensor SpacesQuoConstructor(T, X): TenSpc, Any → TenSpc, Mapquo< T | X >: TenSpc, Any → TenSpc, MapT / S: TenSpc, TenSpc → TenSpc, MapExample: Quotient Tensor Spaces
- Homotopisms
- Constructions of Homotopisms
Homotopism(T, S, M : parameters): TenSpcElt, TenSpcElt, List → HmtpHomotopism(T, S, M : parameters): TenSpcElt, TenSpcElt, SeqEnum → HmtpHomotopism(T, S, M, C : parameters): TenSpcElt, TenSpcElt, List, TenCat → HmtpHomotopism(T, S, M, C : parameters): TenSpcElt, TenSpcElt, SeqEnum, TenCat → HmtpHomotopism(M, C): List, TenCat → HmtpHomotopism(M, C): SeqEnum, TenCat → HmtpIsHomotopism(T, s, H): TenSpcElt, TenSpcElt, Hmtp → BoolEltIsHomotopism(T, s, M): TenSpcElt, TenSpcElt, List → BoolEltIsHomotopism(T, s, M): TenSpcElt, TenSpcElt, SeqEnum → BoolEltIsHomotopism(T, s, M, C): TenSpcElt, TenSpcElt, List, TenCat → BoolEltIsHomotopism(T, s, M, C): TenSpcElt, TenSpcElt, SeqEnum, TenCat → BoolEltExample: Homotopism ConstExample: Mixed Homotopisms
- Basic Operations with Homotopisms
H1 * H2: Hmtp, Hmtp → HmtpH . a: Hmtp, RngIntElt → MapExample: Homotopism OpsPrecompose(T, f, a): TenSpcElt, Map, RngIntElt → TenSpcEltPrecompose(T, M, a): TenSpcElt, Mtrx, RngIntElt → TenSpcEltT @ H: TenSpcElt, Hmtp → TenSpcElt
- Basic Properties of Homotopisms
- Linear Invariants of Tensors
- Invariants for Bilinear Tensors
- Invariants of General Multilinear Maps
Centroid(T): TenSpcElt → AlgMatCentroid(T, A): TenSpcElt, {RngIntElt} → AlgMatExample: CentroidDerivationAlgebra(T): TenSpcElt → AlgMatLieDerivationAlgebra(T, A): TenSpcElt, {RngIntElt} → AlgMatLieNucleus(T, a, b): TenSpcElt, RngIntElt, RngIntElt → AlgMatNucleus(T, A): TenSpcElt, SetEnum → AlgMatExample: Restrict DerivationSelfAdjointAlgebra(t, a, b): TenSpcElt, RngIntElt, RngIntElt → ModMatFldTensorOverCentroid(T): TenSpcElt → TenSpcElt, HmtpExample: Centroid Unipotent
- Some Extended Examples
