transpose
The transpose 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
A=(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn) |
then
At=(a11a21⋯am1a12a22⋯am2⋮⋮⋱⋮a1na2n⋯anm) |
Note that the transpose of an m×n matrix is a n×m matrix.
Properties
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 vectors with n rows. Then
-
1.
(Ct)t=C
-
2.
(C+D)t=Ct+Dt
-
3.
(cD)t=cDt
-
4.
(DE)t=EtDt.
-
5.
(AB)t=BtAt.
-
6.
If A is invertible
, then (At)-1=(A-1)t
-
7.
If A is real, trace(AtA)≥0 (where trace is the trace of a matrix).
-
8.
The transpose is a linear mapping from the vector space
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 product can also be defined using the matrix transpose. If x and y are column vectors with n rows each,
xty=x⋅y |
which implies
xtx=x⋅x=|| |
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 |