identity matrix
The n×n identity matrix I (or In) over a ring R (with an identity
1) is the square matrix
with coefficients in R given by
I=[10⋯001⋯000⋱000⋯1], |
where the numeral “1” and “0” respectively represent the multiplicative and additive identities in R.
0.0.1 Properties
The identity matrix In serves as the multiplicative identity in the ring of n×n matrices over R with standard matrix multiplication. For any n×n matrix M, we have InM=MIn=M, and the identity matrix is uniquely defined by this property. In addition
, for any n×m matrix A and m×n B, we have IA=A and BI=B.
The n×n identity matrix I satisfy the following properties
-
•
For the determinant
, we have , and for the trace, we have .
-
•
The identity matrix has only one eigenvalue
of multiplicity . The corresponding eigenvectors
can be chosen to be .
-
•
The matrix exponential
of gives .
-
•
The identity matrix is a diagonal matrix
.
Title | identity matrix |
---|---|
Canonical name | IdentityMatrix |
Date of creation | 2013-03-22 12:06:29 |
Last modified on | 2013-03-22 12:06:29 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 13 |
Author | mathcam (2727) |
Entry type | Definition |
Classification | msc 15-01 |
Classification | msc 15A57 |
Related topic | KroneckerDelta |
Related topic | ZeroMatrix |
Related topic | IdentityMap |