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is a ring under entrywise addition and matrix multiplication, including the zero ring
Statement
For every field and natural , is a ring under entrywise addition and matrix multiplication, with zero matrix as additive identity and as multiplicative identity. For , this is the one-element zero ring .
Facts & Assumptions
Given: A field and a natural .
Matrix multiplication is associative and unital and distributes over entrywise addition on both sides (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).
Proof
The published pointwise vector-space structure on makes entrywise addition an abelian group operation, with the zero matrix and entrywise negatives.
By [L1], matrix multiplication is associative, has identity , and satisfies both distributive laws over that addition.
These are exactly the ring axioms. If , there is one empty matrix, so its zero and identity coincide; the ring convention permits , making the zero ring.
Depends on
- The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations
- Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication
- Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
Used by
- End_F(V) is a ring and matrix representation is a ring isomorphism End_F(V)≅ Mₙ(F) Corollary
- GLₙ(F) is a group under matrix multiplication, including the trivial group GL₀(F) Corollary
- Mₙ(F) is noncommutative for every n≥2 Corollary
- Invertible matrices and the general linear group GLₙ(F) Definition
- For a field extension K/F, one has K⊗_FMₙ(F)≅ Mₙ(K) as K-algebras Example
- Determinant criterion: if the matrix (⟨ vᵢ,wⱼ⟩)_i,j is invertible then both v₁,…,vₘ and w₁,…,wₘ are linearly independent Lemma
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., §3C (standard reference, not scraped)