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

then

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 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