矩阵-转置矩阵¶

定义¶

如果:

\begin{array}{r} A = [\begin{array}{cccc} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{m 1} & a_{m 2} & \dots & a_{m n} \end{array}] \end{array}

那么其转置矩阵 \mathbf{A}^{\top} 为:

\begin{array}{r} A^{⊤} = [\begin{array}{cccc} a_{11} & a_{21} & \dots & a_{m 1} \\ a_{12} & a_{22} & \dots & a_{m 2} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ a_{1 n} & a_{2 n} & \dots & a_{m n} \end{array}] \end{array}

\begin{array}{r} 如 果 A \in R^{m \times n} \\ 则 A^{⊤} \in R^{n \times m} \end{array}

(A^{⊤})^{⊤} = A

{A + B}^{⊤} = A^{⊤} + B^{⊤}

(c A)^{⊤} = c B^{⊤}

(AB)^{⊤} = B^{⊤} A^{⊤}

\begin{array}{r} 如 果 A^{⊤} = A \\ 则 A 是 对 称 矩 阵 \end{array}