# 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.