Transpose

(Redirected from Transpose Matrix)

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 AT,[1][2] A[3] Atr and At.

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:[3]

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

Transpose Media

Related pages

References

  1. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-08.
  2. Nykamp, Duane. "The transpose of a matrix". Math Insight. Retrieved September 8, 2020.
  3. 3.0 3.1 Weisstein, Eric W. "Transpose". mathworld.wolfram.com. Retrieved 2020-09-08.