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.

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

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,

which implies

which is another way of defining the square of the vector Euclidean norm.