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 ${A}^{t}$, although sometimes it is written as ${A}^{T}$ or ${A}^{\prime}$. So if $A$ is an $m\times n$ matrix and
$$A=\left(\begin{array}{cccc}\hfill {a}_{11}\hfill & \hfill {a}_{12}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{1n}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{2n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{m1}\hfill & \hfill {a}_{m2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{mn}\hfill \end{array}\right)$$ 
then
$${A}^{t}=\left(\begin{array}{cccc}\hfill {a}_{11}\hfill & \hfill {a}_{21}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{m1}\hfill \\ \hfill {a}_{12}\hfill & \hfill {a}_{22}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{m2}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{1n}\hfill & \hfill {a}_{2n}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{nm}\hfill \end{array}\right)$$ 
Note that the transpose of an $m\times n$ matrix is a $n\times m$ matrix.
Properties
Let $A$ and $B$ be $m\times m$ matrices, $C$ and $D$ be $m\times n$ matrices, $E$ be an $n\times k$ matrix, and $c$ be a constant. Let $x$ and $y$ be column vectors^{} with $n$ rows. Then

1.
${({C}^{t})}^{t}=C$

2.
${(C+D)}^{t}={C}^{t}+{D}^{t}$

3.
${(cD)}^{t}=c{D}^{t}$

4.
${(DE)}^{t}={E}^{t}{D}^{t}.$

5.
${(AB)}^{t}={B}^{t}{A}^{t}.$

6.
If $A$ is invertible^{} , then ${({A}^{t})}^{1}={({A}^{1})}^{t}$

7.
If $A$ is real, $\mathrm{trace}({A}^{t}A)\ge 0$ (where $\mathrm{trace}$ is the trace of a matrix).

8.
The transpose is a linear mapping from the vector space^{} of matrices to itself. That is, ${(\alpha A+\beta B)}^{t}=\alpha {(A)}^{t}+\beta {(B)}^{t}$, for samesized matrices $A$ and $B$ and scalars $\alpha $ and $\beta $.
The familiar vector dot product^{} can also be defined using the matrix transpose. If $x$ and $y$ are column vectors with $n$ rows each,
$${x}^{t}y=x\cdot y$$ 
which implies
$${x}^{t}x=x\cdot x={x}_{2}^{2}$$ 
which is another way of defining the square of the vector Euclidean norm.
Title  transpose 
Canonical name  Transpose 
Date of creation  20130322 12:01:02 
Last modified on  20130322 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 