Transpose

This page 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, A′ A^tr and A^t.

Examples

Here is the vector [math]\displaystyle{ \begin{bmatrix} 1 & 2 \end{bmatrix} }[/math] being transposed:

• [math]\displaystyle{ \begin{bmatrix} 1 & 2 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 \\ 2 \end{bmatrix}. }[/math]

Here are a few matrices being transposed:

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

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

• [math]\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}. \; }[/math]

Properties

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

• [math]\displaystyle{ (A^T)^{-1} = (A^{-1})^T }[/math]
• [math]\displaystyle{ (AB)^T = B^T A^T }[/math]

Transpose Media

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  The transpose AT of a matrix A can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.

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  Illustration of row- and column-major order

Related pages

• Invertible matrix