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

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The transpose $A^{\mathsf T}$ of a matrix

Definition

For $A=(a_{ij})\in M_{m\times n}(F)$ , the transpose of $A$ is the matrix $A^{\mathsf T}\in M_{n\times m}(F)$ defined by

$(A^{\mathsf T})_{ji}:=a_{ij}.$

Thus transposition interchanges rows and columns. The transpose of a zero-sized matrix is the unique matrix of the transposed zero-sized 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

S. Axler, Linear Algebra Done Right, 4th ed., §3C, Exercises 14–15 (standard reference, not scraped)

The transpose ATA^{\mathsf T}AT of a matrix