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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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 38 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- S. Axler, Linear Algebra Done Right, 4th ed., §3C (standard reference, not scraped)