Definition:Transpose of Matrix

Definition

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over a set.

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

$\mathbf A^\intercal = \sqbrk \beta_{n m}: \forall i \in \closedint 1 n, j \in \closedint 1 m: \beta_{i j} = \alpha_{j i}$

Also denoted as

The transpose of a matrix is often seen indicated by a lowercase or uppercase $\text T$:

$\mathbf A^t$

$\mathbf A^T$ or $\mathbf A^{\mathrm T}$

$^t\!\mathbf A$

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

Also see

• Results about transposes of matrices can be found here.

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): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): transpose: 2, 3.
• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.4$ The transpose of a matrix
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): transpose
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): transpose
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): transpose

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