查看“︁空矩阵”︁的源代码

{{Other uses
|subject = 行数或列数为零的[[矩陣]]
|other = 行数與列数皆不为零，但其元素全為零的矩陣
|零矩陣
}}
'''空矩阵'''是指至少有一個[[向量空间的维数|維度]]為零的[[矩陣]]，亦即行数或列数为零的矩阵。<ref>{{Cite web | url=http://omatrix.com/manual/glossary.htm | archive-url=https://web.archive.org/web/20090429015728/http://omatrix.com/manual/glossary.htm | archive-date=2009-04-29 | title="Empty Matrix: A matrix is empty if either its row or column dimension is zero"|publisher=O-Matrix v6 User Guide}}</ref><ref>{{Cite web | url=http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html | title=Matrix - MATLAB Data Structures|publisher=system.nada.kth.se|quote=A matrix having at least one dimension equal to zero is called an empty matrix | archive-url=https://web.archive.org/web/20091228102653/http://www.system.nada.kth.se/unix/software/matlab/Release_14.1/techdoc/matlab_prog/ch_dat29.html | archive-date=2009-12-28}}</ref>最小的空矩陣為0×0矩陣。空矩陣亦可以是0×5或10×0等形式<ref>{{Cite web | url=http://www.ece.northwestern.edu/local-apps/matlabhelp/techdoc/matlab_prog/ch10_p44.html | title=Empty Matrices | publisher=www.ece.northwestern.edu | access-date=2022-04-29 | archive-date=2020-02-18 | archive-url=https://web.archive.org/web/20200218103022/http://www.ece.northwestern.edu/local-apps/matlabhelp/techdoc/matlab_prog/ch10_p44.html }}</ref>。空矩陣不會存在任何元素。

== 定義 ==
空矩阵的定义可以完善一些关于[[零维空间]]的约定。包括约定一个矩阵与空矩阵[[矩陣乘法|相乘]]得到的也是空矩阵，两个<math>n \times 0</math>和<math>0 \times p</math>的空矩阵相乘是一个<math>n \times p</math>的[[零矩阵]]（所有元素都是零的矩阵）。0×0的空矩阵的行列式约定为1，所以它也可以有逆矩阵，约定为它自己<ref name="Faliva & Zoia 2008">{{Citation |last1= Faliva |first1= Mario  |last2= Zoia  |first2= Maria Grazia |title=Dynamic Model Analysis: Advanced Matrix Methods and Unit-Root Econometrics Representation Theorems|publisher=Springer-Verlag |location=Berlin, DE; New York, NY |edition=2nd |isbn= 9783540859956  |year=2008 |page=218}}</ref>{{rp|18}}。

== 性質 ==
*維數相同的空矩阵與空矩阵相乘仍為空矩阵<ref name="4.1.1 Empty Matrices">{{cite web | url = https://octave.org/doc/v4.2.1/Empty-Matrices.html | title = 4.1.1 Empty Matrices | publisher = octave.org | access-date = 2022-04-29 | archive-date = 2019-09-13 | archive-url = https://web.archive.org/web/20190913212543/http://octave.org/doc/v4.2.1/Empty-Matrices.html }}</ref>
*:<math>\left[\ \right]\times\left[\ \right]=\left[\ \right]</math>
*空矩阵與純量或向量相乘仍為空矩阵<ref name="4.1.1 Empty Matrices"/>
*<math>n \times 0</math>的空矩阵和<math>0 \times p</math>的空矩阵相乘結果為<math>n \times p</math>的[[零矩阵]]<ref name="4.1.1 Empty Matrices"/>
*<math>0 \times m</math>的空矩阵和任一<math>m \times n</math>的矩阵相乘結果為<math>0 \times n</math>的空矩阵<ref name="4.1.1 Empty Matrices"/>
*任一<math>m \times n</math>的矩阵和<math>n \times 0</math>的空矩阵相乘結果為<math>m \times 0</math>的空矩阵<ref name="4.1.1 Empty Matrices"/>
*空矩阵的行列式约定为1，即[[空積]]。<ref name="Faliva & Zoia 2008"/>
*空矩阵<math>\left[\ \right]</math>等於零維[[零矩陣]]<math>\mathbf{0}_{0\times 0}</math>等於零維[[單位矩陣]]<math>I_{0\times 0}</math>。<ref name="IEEE277245">{{Cite journal | author=Nett, C.N. and Haddad, W.M.|journal=IEEE Transactions on Automatic Control|title=A system-theoretic appropriate realization of the empty matrix concept|year=1993|volume=38|number=5|pages=771-775|doi=10.1109/9.277245}}</ref>
*:<math>\left[\ \right]=\mathbf{0}_{0\times 0}=I_{0\times 0}=\left[\ \right]^{-1}</math>
*空矩阵的反矩陣為自身。<ref name="Faliva & Zoia 2008"/>{{rp|18}}
*:由於<math>\left[\ \right]=I_{0\times 0}=\left[\ \right]^{-1}</math>
*:因此<math>\left[\ \right] \left[\ \right]^{-1}=\left[\ \right]=I_{0\times 0}</math>，滿足反矩陣與自身相乘為[[單位矩陣]]的定義。
*空矩阵的[[秩 (线性代数)|秩]]為0<ref name="scilab">{{cite web | url = https://help.scilab.org/docs/6.0.0/en_US/empty.html | title = empty matrix | publisher = scilab.org | access-date = 2022-04-29 | archive-date = 2020-12-05 | archive-url = https://web.archive.org/web/20201205005125/https://help.scilab.org/docs/6.0.0/en_US/empty.html }}</ref>

== 參見 ==
*[[零矩陣]]

== 參考文獻 ==
{{Reflist}}

[[Category:矩陣]]
[[Category:零]]