Transpose

This article is about the transpose of a matrix. For other uses, see Transposition (disambiguation).

The transpose of a matrix A is another matrix where the rows of A are written as columns. Vectors can be transposed in the same way. We can write the transpose of A using different symbols such as A^T,^[1][2] A′ ^[3] A^tr and A^t.

Examples

Here is the vector ${\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}}$ being transposed:

• ${\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1\\2\end{bmatrix}}.}$

Here are a few matrices being transposed:

• ${\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3\\2&4\end{bmatrix}}.}$

• ${\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}.\;}$

• ${\displaystyle {\begin{bmatrix}1&2&8\\3&4&3\\5&6&1\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\\8&3&1\end{bmatrix}}.\;}$

Properties

Given two matrices A and B, the following properties related to the transpose are true:^[3]

• ${\displaystyle (A^{T})^{-1}=(A^{-1})^{T}}$
• ${\displaystyle (AB)^{T}=B^{T}A^{T}}$

Related pages

• Invertible matrix