Matrix Transposition

Definition: Matrix Transposition

The transpose of a matrix $A\in F^{m\times n}$ is the matrix $A^{\mathsf{T}}\in F^{n\times m}$ obtained by switching the rows and the columns of $A$, i.e. the $i$-th row (or column) of $A$ is the $i$-th column (or row) of $A^{\mathsf{T}}$:

$${\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]}^{\mathsf{T}}\stackrel{\text{def}}{=}\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{m1}\\ ⋮& \ddots & ⋮\\ a_{1n}& \cdots & a_{mn}\end{array}\right]$$

TIP

The entry in the $i$-th row and the $j$-th column of $A$ is the entry in the $j$-th row and the $i$-th column of $A^{\mathsf{T}}$.

TIP

The number of rows in $A^{\mathsf{T}}$ is equal to the number columns in $A$ and the number of columns in $A^{\mathsf{T}}$ is equal to the number of rows in $A$.

Theorem: Distributivity of Transposition

The matrix transposition is distributive over matrix addition and scalar multiplication:

$$(A+B{)}^{\mathsf{T}}=A^{\mathsf{T}}+B^{\mathsf{T}}$$ $$(\lambda A{)}^{\mathsf{T}}=\lambda A^{\mathsf{T}}$$

PROOF

TODO

Theorem: Antidistributivity of Transposition

The matrix transposition is antidistributive over the matrix product:

$$(AB{)}^{\mathsf{T}}=B^{\mathsf{T}}{A}^{\mathsf{T}}$$

PROOF

TODO

Theorem: Involution of Transposition

The matrix transposition is an involution.

PROOF

TODO