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

 $A=\begin{pmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{pmatrix}$

then

 $A^{t}=\begin{pmatrix}a_{11}&a_{21}&\cdots&a_{m1}\\ a_{12}&a_{22}&\cdots&a_{m2}\\ \vdots&\vdots&\ddots&\vdots\\ a_{1n}&a_{2n}&\cdots&a_{nm}\end{pmatrix}$

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. 1.

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

2. 2.

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

3. 3.

$(cD)^{t}=cD^{t}$

4. 4.

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

5. 5.

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

6. 6.
7. 7.

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

8. 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 same-sized 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 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