Matrix multiplication

One operation, many viewpoints

This chapter builds matrix multiplication from the column-wise definition, derives common computation rules, connects the product AB to composition of linear transformations and develops structural consequences (block multiplication, transpose rule and rank inequalities).

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Chapter contents

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Topic	Pages
From: column-wise multiplication To: element-wise multiplication	1–2
From: element-wise multiplication To: row-wise multiplication	3–4
From: element-wise multiplication To: sum of outer products	5
The 4 ways of matrix-matrix multiplication	6–7
Associativity of matrix multiplication: (AB)C = A(BC)	8–9
Other algebraic properties of matrix multiplication	10–11
Matrix multiplication as composition of linear transformations	12–13
Composition of linear transformations vs composition of algebraic functions	14
Matrix product shapes	15–16
Block multiplication, 2×2 form	17–18
Other valid forms of block multiplication	19–20
Transpose of matrix product (AB)ᵀ	21–22
Products AD & DA (D = diagonal matrix)	23–24
Product of orthonormal matrices PQ	25
Product of upper triangular matrices TU	26–28
rank(AB) ≤ min(rank(A), rank(B))	29
rank(AᵀA) = rank(A)	30–31

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