zero matrix

The $n\times m$ zero $O$ 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 (http://planetmath.org/Ring) in $R$ .

0.0.1 Properties

The zero matrix is the additive identity in the ring of $n\times n$ matrices over $R$ . This is an alternative definition of $O$ (since there’s just one additive identity in any given ring (http://planetmath.org/UniquenessOfAdditiveIdentityInARing2)).

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.

Title	zero matrix
Canonical name	ZeroMatrix
Date of creation	2013-03-22 14:19:19
Last modified on	2013-03-22 14:19:19
Owner	waj (4416)
Last modified by	waj (4416)
Numerical id	8
Author	waj (4416)
Entry type	Definition
Classification	msc 15-01
Related topic	Matrix
Related topic	IdentityMatrix