Theorems · Theorem · linear algebra

TensorProduct.toMatrix_assoc

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} {ι : Type u_7} {κ : Type u_8} {τ : Type u_9}
  [inst : DecidableEq ι] [inst_1 : DecidableEq κ] [inst_2 : DecidableEq τ] [inst_3 : Fintype ι] [inst_4 : Fintype κ]
  [inst_5 : Fintype τ] [inst_6 : CommRing R] [inst_7 : AddCommGroup M] [inst_8 : AddCommGroup N]
  [inst_9 : AddCommGroup P] [inst_10 : Module R M] [inst_11 : Module R N] [inst_12 : Module R P]
  (bM : Module.Basis ι R M) (bN : Module.Basis κ R N) (bP : Module.Basis τ R P),
  (LinearMap.toMatrix ((bM.tensorProduct bN).tensorProduct bP) (bM.tensorProduct (bN.tensorProduct bP)))
      ↑(TensorProduct.assoc R M N P) =
    Matrix.submatrix 1 id ⇑(Equiv.prodAssoc ι κ τ)

TensorProduct.assoc corresponds to a permutation of the identity matrix.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Matrix

Cited by

0 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DecidableEqDecidableEqDecidableEqFintypeFintypeFintypeCommRingAddCommGroupAddCommGroupAddCommGroupModuleModuleModule