The transposeMathworldPlanetmath of a matrix A is the matrix formed by “flipping” A about the diagonal line from the upper left corner. It is usually denoted At, although sometimes it is written as AT or A. So if A is an m×n matrix and




Note that the transpose of an m×n matrix is a n×m matrix.


Let A and B be m×m matrices, C and D be m×n matrices, E be an n×k matrix, and c be a constant. Let x and y be column vectorsMathworldPlanetmath with n rows. Then

  1. 1.


  2. 2.


  3. 3.


  4. 4.


  5. 5.


  6. 6.

    If A is invertiblePlanetmathPlanetmath , then (At)-1=(A-1)t

  7. 7.

    If A is real, trace(AtA)0 (where trace is the trace of a matrix).

  8. 8.

    The transpose is a linear mapping from the vector spaceMathworldPlanetmath of matrices to itself. That is, (αA+βB)t=α(A)t+β(B)t, for same-sized matrices A and B and scalars α and β.

The familiar vector dot productMathworldPlanetmath can also be defined using the matrix transpose. If x and y are column vectors with n rows each,


which implies


which is another way of defining the square of the vector Euclidean norm.

Title transpose
Canonical name Transpose
Date of creation 2013-03-22 12:01:02
Last modified on 2013-03-22 12:01:02
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 12
Author mathcam (2727)
Entry type Definition
Classification msc 15A57
Related topic AdjointEndomorphism
Related topic HermitianConjugate
Related topic FrobeniusMatrixNorm
Related topic ConjugateTranspose
Related topic TransposeOperator
Related topic VectorizationOfMatrix