identity matrix (original) (raw)

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 In⁢M=M⁢In=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 I⁢A=A and B⁢I=B.

The n×n identity matrix I satisfy the following properties

• •
  For the determinant, we have det⁡I=1, and for the trace, we havetr⁡I=n.
• •
  The identity matrix has only one eigenvalue λ=1 ofmultiplicity n. The corresponding eigenvectors can be chosen to bev1=(1,0,…,0),…,vn=(0,…,0,1).
• •
• •