Transpose

For $\mathbf{A}\in {\mathbb{R}}^{m\times n}$ the matrix $\mathbf{B}\in {\mathbb{R}}^{n\times m}$ with $b_{ij}=a_{ji}$ is called the transpose of $\mathbf{A}$. We write $\mathbf{B}={\mathbf{A}}^{\top }$.

Some Properties

\begin{aligned}\left(\boldsymbol{A}^{\top}\right)^{\top} &=\boldsymbol{A} \\(\boldsymbol{A}+\boldsymbol{B})^{\top} &=\boldsymbol{A}^{\top}+\boldsymbol{B}^{\top} \\ (\alpha \mathbf{A})^{\top} &=\alpha \mathbf{A}^{\top}\\ (\boldsymbol{A} \boldsymbol{B})^{\top} &=\boldsymbol{B}^{\top} \boldsymbol{A}^{\top} \end{aligned}$$ ## With Scalar $(\lambda \boldsymbol{C})^{\top}=\boldsymbol{C}^{\top} \lambda^{\top}=\boldsymbol{C}^{\top} \lambda=\lambda \boldsymbol{C}^{\top}$ since $\lambda=\lambda^{\top}$ for all $\lambda \in \mathbb{R}$ ## With Inverse 만약 $\boldsymbol{A}$ 가 [[Inverse matrix|invertible]] 하면, $\boldsymbol{A}^{\top}$ 도 그렇다. 그래서 $\left(\boldsymbol{A}^{-1}\right)^{\top}=\left(\boldsymbol{A}^{\top}\right)^{-1}=: \boldsymbol{A}^{-\top}$ 를 만족한다. # Related # References