# Definition:Transpose of Matrix

## Definition

Let $\mathbf A = \left[{\alpha}\right]_{m n}$ be an $m \times n$ matrix over a set.

Then the **transpose** of $\mathbf A$ is denoted $\mathbf A^\intercal$ and is defined as:

- $\mathbf A^\intercal = \left[{\beta}\right]_{n m}: \forall i \in \left[{1 \,.\,.\, n}\right], j \in \left[{1 \,.\,.\, m}\right]: \beta_{i j} = \alpha_{j i}$

## Also denoted as

The **transpose** is often seen indicated by a lowercase or uppercase T:

- $\mathbf A^t$
- $\mathbf A^T$
- $^t\!\mathbf A$

but these are usually considered suboptimal in the contemporary technological environment.

## Technical note

The $\LaTeX$ code used to denote $\intercal$ is a superscripted `\intercal`

.

Thus $\mathbf A^\intercal$ is encoded as `\mathbf A^\intercal`

.

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 29$

- For a video presentation of the contents of this page, visit the Khan Academy.