Elementary Properties of the Transpose of a Real Matrix

Let $m$, $n$ and $p$ be natural numbers with $1\le m$, $1\le n$ and $1\le p$, let $\mathbb{R}$ be the real numbers, and for a natural number $q$ let $[q]$ be the initial segment determined by $q$. Let $A$ and $B$ be real $m\times n$ matrices, let $C$ be a real $n\times p$ matrix, and let $\mu\in\mathbb{R}$.

Write $A^{\top}$ for the transpose, $A+B$ for the sum and $\mu A$ for the scalar multiple of matrices, $AC$ for the matrix product, $I_{n}$ for the identity matrix of size $n$, and $Av$ for the matrix-vector product. On Euclidean space write $w\cdot z$ for the dot product.

Then the following hold.

1. (Involution) $(A^{\top})^{\top}=A$.

2. (Linearity) $(A+B)^{\top}=A^{\top}+B^{\top}$ and $(\mu A)^{\top}=\mu\,A^{\top}$.

3. (Identity matrix) $I_{n}^{\top}=I_{n}$.

4. (Products) $(AC)^{\top}=C^{\top}A^{\top}$.

5. (Adjoint relation) For all $v\in\mathbb{R}^{n}$ and $w\in\mathbb{R}^{m}$,

$$w\cdot(Av)=(A^{\top}w)\cdot v .$$

6. (Inverses) Suppose $m=n$ and $A$ is invertible, with inverse $A^{-1}$. Then $A^{\top}$ is invertible and $(A^{\top})^{-1}=(A^{-1})^{\top}$.