# 行向量與列向量

（重定向自列向量

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}\,}$

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\dots &x_{m}\end{bmatrix}}\,}$

${\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}\,}$

${\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}^{\rm {T}}={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}\,}$

## 標示

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}}$

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}}$

Row vector Column vector
Standard matrix notation
(array spaces, no commas, transpose signs)
${\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}{\text{ or }}{\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{m}\end{bmatrix}}^{\rm {T}}}$
Alternative notation 1
(commas, transpose signs)
${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}^{\rm {T}}}$
Alternative notation 2
(commas and semicolons, no transpose signs)
${\displaystyle {\begin{bmatrix}x_{1},x_{2},\dots ,x_{m}\end{bmatrix}}}$ ${\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{m}\end{bmatrix}}}$

## 操作

${\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} ^{\mathrm {T} }\mathbf {b} ={\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}}{\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\,,}$

${\displaystyle \mathbf {b} \cdot \mathbf {a} =\mathbf {b} ^{\mathrm {T} }\mathbf {a} ={\begin{bmatrix}b_{1}&b_{2}&b_{3}\end{bmatrix}}{\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}\,.}$

${\displaystyle \mathbf {a} \otimes \mathbf {b} =\mathbf {a} \mathbf {b} ^{\mathrm {T} }={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\end{bmatrix}}{\begin{bmatrix}b_{1}&b_{2}&b_{3}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}&a_{1}b_{2}&a_{1}b_{3}\\a_{2}b_{1}&a_{2}b_{2}&a_{2}b_{3}\\a_{3}b_{1}&a_{3}b_{2}&a_{3}b_{3}\\\end{bmatrix}}\,,}$

${\displaystyle \mathbf {b} \otimes \mathbf {a} =\mathbf {b} \mathbf {a} ^{\mathrm {T} }={\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\end{bmatrix}}{\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}}={\begin{bmatrix}b_{1}a_{1}&b_{1}a_{2}&b_{1}a_{3}\\b_{2}a_{1}&b_{2}a_{2}&b_{2}a_{3}\\b_{3}a_{1}&b_{3}a_{2}&b_{3}a_{3}\\\end{bmatrix}}\,.}$

## 优选输入的向量矩阵变换

${\displaystyle vM=p\,.}$

${\displaystyle pQ=t\,.}$

${\displaystyle p=Mv\,,\quad t=Qp}$,

${\displaystyle \alpha (ST)=(\alpha S)T=\beta T=\gamma }$中。"

(希腊字母代表行向量)。

## 備註

1. ^ Meyer (2000), p. 8
2. ^ Raiz A. Usmani (1987) Applied Linear Algebra Marcel Dekker ISBN 0824776224.