13 - Transpose

Transpose Matrix of $n\times n$ matrix $A$

Transpose $A^T$ defined such that

$$(i,j)\text{th entry of }{A}^T=(j,i)\text{th entry of A}$$

Essentially reflecting across the main diagonal of the matrix

Properties of Transpose

1. Addition and Scalar Multiplication Rules (for $A,B$ being $m\times n$ matrices and $\lambda \in \mathbb{R}$)

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

2. Product Rule (for $A$ being a $m\times n$ matrix and $B$ being a $n\times p$ matrix)

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

3. Involution Rule (for $A$ being a $m\times n$ matrix)

$$(A^T{)}^T=A$$

4. Inverse Rule ($A$ invertible $⟺$ $A^T$ invertible)

$$(A^T{)}^{-1}=(A^{-1}{)}^T$$