# 零矩陣

${\displaystyle O_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ O_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ O_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}}}$

## 性質

• ${\displaystyle m\times n}$ 的零矩陣${\displaystyle O}$ ${\displaystyle m\times n}$ 的任意矩陣${\displaystyle A}$ 的和為${\displaystyle A+O=O+A=A}$ ，差為${\displaystyle A-O=A}$ ${\displaystyle O-A=-A}$
• ${\displaystyle l\times m}$ 的零矩陣${\displaystyle O}$ ${\displaystyle m\times n}$ 的任意矩陣${\displaystyle A}$ 的積${\displaystyle OA}$ ${\displaystyle l\times n}$ 的零矩陣。
• ${\displaystyle l\times m}$ 的任意矩陣${\displaystyle B}$ ${\displaystyle m\times n}$ 的零矩陣${\displaystyle O}$ 的積${\displaystyle BO}$ ${\displaystyle l\times n}$ 的零矩陣。

## 參考文獻

1. ^ Bronson, Richard; Costa, Gabriel B., Linear Algebra: An Introduction, Academic Press: 377, 2007, ISBN 9780120887842, The zero matrix represents the zero transformation 0, having the property 0(v) = 0 for every vector v ∈ V.