DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes

Definition

Let $F$ be a field, let $m, n, p \in N$ , and let $A = (a_{i j}) \in M_{m \times n} (F)$ and $B = (b_{j k}) \in M_{n \times p} (F)$ . Their matrix product is the matrix $A B \in M_{m \times p} (F)$ whose entries are

$(A B)_{i k} : = \sum_{j \in n} a_{i j} b_{j k} .$

The sum is the finite sum in the additive commutative monoid of $F$ . If $n = 0$ , it is the empty sum, so $A B$ is the zero matrix of shape $m$ by $p$ .

The identity matrix $I_{n} \in M_{n} (F)$ has entry $1$ when $i = j$ and entry $0$ when $i \neq j$ . In particular, $I_{0}$ is the unique empty matrix in $M_{0} (F)$ .

Depends on

Dependency tree · 2 levels

Sources

S. Axler, Linear Algebra Done Right, 4th ed., §3C

Rectangular matrix multiplication and the identity matrix InIn​, including zero-sized shapes

Definition

Depends on

Dependency tree · 2 levels

Sources

Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes