Matrix Transpose

linear-algebra

Definition

Matrix Transpose

Let $A\in F^{m\times n}$ be a matrix over a field $F$.

The transpose of $A$, written $A^{\top }$, is the matrix obtained by swapping row and column elements.

Einstein Notation: $(A^{\top }{)}_{ij}=A_{ji}$.

Properties

Let $A$ and $B$ be two matrices.

Idempotent

$$(A^{\top }{)}^{\top }=A$$

Addition

$$(A+B{)}^{\top }=A^{\top }+B^{\top }$$

Scalar Multiplication

$$(c\cdot A{)}^{\top }=c\cdot A^{\top }$$