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Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication

Statement

Matrix multiplication over a field is associative and unital, distributes over entrywise addition on both sides, and is compatible with scalar multiplication. Precisely, whenever the displayed products have compatible shapes,

$A(BC)=(AB)C,\qquad I_mA=A=AI_n,$

$(A+A')B=AB+A'B,\qquad A(B+B')=AB+AB',$

$0B=0=A0,\qquad (\lambda A)B=\lambda(AB)=A(\lambda B).$

These identities include every zero-sized shape.

Facts & Assumptions

Given: A field $F$ ; matrices $A\in M_{m\times n}(F)$ , $B\in M_{n\times p}(F)$ , and $C\in M_{p\times q}(F)$ ; conformable matrices $A',B'$ ; and a scalar $\lambda\in F$ .

[L1]

Matrix products and identity matrices are defined entrywise by finite row-by-column sums (Rectangular matrix multiplication and the identity matrix $I_n$ , including zero-sized shapes).

[L2]

A finite double sum may be evaluated in either order (Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule).

Proof

technique · direct

1.1

For $i<m$ and $\ell<q$ , expansion by [L1] gives $((AB)C)_{i\ell}=\sum_{k<p}\sum_{j<n}(a_{ij}b_{jk})c_{k\ell}$ and $(A(BC))_{i\ell}=\sum_{j<n}\sum_{k<p}a_{ij}(b_{jk}c_{k\ell})$ .

givenL1

2.1

Associativity and distributivity in $F$ , followed by [L2], identify the two sums in step 1.1, so $(AB)C=A(BC)$ .

step 1.1L1L2

3.1

The entry formulas also give $(I_mA)_{ik}=a_{ik}=(AI_n)_{ik}$ ; splitting sums gives both distributive laws; pulling out $\lambda$ gives scalar compatibility; and a sum of zero summands gives each zero law. Thus all stated identities hold entrywise.

step 2.1L1

4.1

If any indexing dimension is zero, every asserted matrix equality either has no entries to compare or reduces to an empty sum, whose value is $0$ . Hence the same calculations cover all zero-sized shapes, including $I_0$ .

step 3.1L1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 39 results over 11 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)