DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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Matrix units $E_{ij}$ and the Kronecker delta

Definition

For indices $a,b$ in the same finite index set, the Kronecker delta is

$\delta_{ab}:=\begin{cases}1,&a=b,\\0,&a\ne b.\end{cases}$

Let $i<m$ and $j<n$ . The matrix unit $E_{ij}\in M_{m\times n}(F)$ is the matrix whose $(r,s)$ -entry is

$(E_{ij})_{rs}:=\delta_{ri}\delta_{sj}.$

Thus $E_{ij}$ has entry $1$ in position $(i,j)$ and entry $0$ everywhere else. When a dimension is zero, no index of that dimension exists and there are no matrix units of the corresponding shape.

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

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 10 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

P. E. Leonard, Linear Algebra Notes, §3.4 (standard reference, not scraped)

Matrix units EijE_{ij}Eij​ and the Kronecker delta