Transpose

The transpose of a matrix A is what you get when you rewrite the matrix with its rows as columns. Vectors can be transposed in the same way.

We can write the transpose of A using different symbols:

• A^T
• A′
• A^tr
• 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]

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