Theorems · Definition · linear algebra

Matrix.reindexAlgEquiv

{m : Type u_2} →
  {n : Type u_3} →
    (R : Type u_11) →
      (A : Type u_12) →
        [inst : CommSemiring R] →
          [inst_1 : Fintype n] →
            [inst_2 : Fintype m] →
              [inst_3 : DecidableEq m] →
                [inst_4 : DecidableEq n] →
                  [inst_5 : Semiring A] → [inst_6 : Algebra R A] → m ≃ n → Matrix m m A ≃ₐ[R] Matrix n n A

For square matrices with coefficients in an algebra over a commutative semiring, the natural map that reindexes a matrix's rows and columns with equivalent types, Matrix.reindex, is an equivalence of algebras.

Defined in

Mathlib.LinearAlgebra.Matrix.Reindex

Cited by

19 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringFintypeFintypeDecidableEqDecidableEqSemiringAlgebra