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 $A'$ . 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. $(C^t)^t = C$
2. $(C+D)^t = C^t + D^t$
3. $(cD)^t = cD^t$
4. $(DE)^t=E^tD^t.$
5. $(AB)^t = B^t A^t.$
6. If $A$ is invertible , then $(A^t)^{-1} = (A^{-1})^t $
7. If $A$ is real, $\operatorname{trace}(A^tA) \ge 0$ (where $\operatorname{trace}$ 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.