zero matrix
The n×m zero O over a ring R is the n×m matrix with coefficients in R given by
O=[0⋯0⋮⋱⋮0⋯0], |
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×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×n zero matrix O has the following properties:
-
•
The determinant
of O is , and its trace is .
-
•
has only one eigenvalue
of multiplicity . Any non-zero vector is an eigenvector
of , so if we’re looking for a basis of eigenvectors, we could pick the standard basis .
-
•
The matrix exponential
of is , the 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 |