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 of \(A\) is the \(i\)-th column of \(A^\mathsf{T}\) and vice versa:

\[ \begin{bmatrix}a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn}\end{bmatrix}^\mathsf{T} \overset{\text{def}}{=} \begin{bmatrix}a_{11} & \cdots & a_{m1} \\ \vdots & \ddots & \vdots \\ a_{1n} & \cdots & a_{mn}\end{bmatrix} \]

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

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

Matrix transposition is antidistributive over matrix products:

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

Proof

TODO