Matrix/K/Transposed matrix/Definition

Transposed matrix

Let ${}K$ be a field, and let ${}M=(a_{ij})_{ij}$ be an ${}m\times n$-matrix over ${}K$. Then the ${}n\times m$-matrix

${M^{\text{tr}}}={\left(b_{ij}\right)}_{ij}{\text{ with }}b_{ij}:=a_{ji}$

is called the transposed matrix for ${}M$.