# 转置矩阵

${\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}$

• A的横行写为AT的纵列
• A的纵列写为AT的横行

${\displaystyle A_{ij}^{\mathrm {T} }=A_{ji}}$ for ${\displaystyle 1\leq i\leq n,}$ ${\displaystyle 1\leq j\leq m}$

## 例子

• ${\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3\\2&4\end{bmatrix}}}$
• ${\displaystyle {\begin{bmatrix}1&2\\3&4\\5&6\end{bmatrix}}^{\mathrm {T} }\!\!\;\!=\,{\begin{bmatrix}1&3&5\\2&4&6\end{bmatrix}}\;}$

## 性质

• ${\displaystyle \left(A^{\mathrm {T} }\right)^{\mathrm {T} }=A\quad }$

• ${\displaystyle (A+B)^{\mathrm {T} }=A^{\mathrm {T} }+B^{\mathrm {T} }}$

• ${\displaystyle \left(AB\right)^{\mathrm {T} }=B^{\mathrm {T} }A^{\mathrm {T} }}$

• ${\displaystyle (cA)^{\mathrm {T} }=cA^{\mathrm {T} }}$

• ${\displaystyle \det(A^{\mathrm {T} })=\det(A)}$

• 两个纵列向量ab点积可计算为
${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathrm {T} }\mathbf {b} ,}$

## 特殊转置矩阵

${\displaystyle A^{\mathrm {T} }=A}$

${\displaystyle GG^{\mathrm {T} }=G^{\mathrm {T} }G=I_{n},\,}$ I单位矩阵

${\displaystyle A^{\mathrm {T} }=-A}$

${\displaystyle A^{H}=({\overline {A}})^{\mathrm {T} }={\overline {(A^{\mathrm {T} })}}}$

## 线性映射的转置

${\displaystyle B_{V}(v,{}^{t}f(w))=B_{W}(f(v),w)\quad \forall \ v\in V,w\in W}$