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 ZeroMatrix 2013-03-22 14:19:19 2013-03-22 14:19:19 waj (4416) waj (4416) 8 waj (4416) Definition msc 15-01 Matrix IdentityMatrix
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