kidzsearch.com > wiki   Explore:web images videos games  

Transpose




KidzSearch Safe Wikipedia for Kids.
Jump to: navigation, search

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]\begin{bmatrix} 1 & 2 \end{bmatrix}[/math] being transposed:

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

Here are a few matrices being transposed:

  • [math]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}. [/math]
  • [math] \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}^{\mathrm{T}} \!\! \;\! = \, \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix}. \; [/math]
  • [math] \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](A^T)^{-1} = (A^{-1})^T[/math]
  • [math](AB)^T = B^T A^T[/math]

Related pages

References

  1. "Comprehensive List of Algebra Symbols" (in en-US). 2020-03-25. https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/. 
  2. Nykamp, Duane. "The transpose of a matrix". https://mathinsight.org/matrix_transpose. 
  3. 3.0 3.1 Weisstein, Eric W.. "Transpose" (in en). https://mathworld.wolfram.com/Transpose.html. 
de:Matrix (Mathematik)#Die transponierte Matrix