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The $n \times m$ zero matrix $O$ over a ring $R$ is the $n \times m$ matrix with coefficients in $R$ given by
$$ O = \begin{bmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ \end{bmatrix},$$
where 0 is the additive identity in $R$ .
The zero matrix is the additive identity in the ring of $n\times n$ matrices over $R$ . This property is an alternative definition of $O$ (since there's just one additive identity in any given ring).
The $n\times n$ zero matrix $O$ has the following properties:
- The determinant of $O$ is $\det O = 0$ , and its trace is $\operatorname{tr}O = 0$ .
- $O$ has only one eigenvalue $\lambda =0$ of multiplicity $n$ . Any non-zero vector is an eigenvector of $O$ , so if we're looking for a basis of eigenvectors, we could pick the standard basis
$e_1=(1,0,\ldots, 0), \ldots , e_n=(0,\ldots, 0,1)$ .
- The matrix exponential of $O$ is $e^O = I$ , the $n\times n$ identity matrix.
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Cross-references: identity matrix, matrix exponential, standard basis, eigenvectors, basis, eigenvector, non-zero vector, multiplicity, eigenvalue, trace, determinant, properties, coefficients, matrix, ring
There are 8 references to this entry.
This is version 5 of zero matrix, born on 2004-04-21, modified 2004-04-25.
Object id is 5789, canonical name is ZeroMatrix.
Accessed 5066 times total.
Classification:
| AMS MSC: | 15-01 (Linear and multilinear algebra; matrix theory :: Instructional exposition ) |
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Pending Errata and Addenda
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