# identity matrix

The $n\times n$ identity matrix $I$ (or $I_{n}$) over a ring $R$ (with an identity 1) is the square matrix with coefficients in $R$ given by

 $I=\begin{bmatrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ 0&0&\ddots&0\\ 0&0&\cdots&1\end{bmatrix},$

where the numeral “1” and “0” respectively represent the multiplicative and additive identities in $R$.

## 0.0.1 Properties

The identity matrix $I_{n}$ serves as the multiplicative identity in the ring of $n\times n$ matrices over $R$ with standard matrix multiplication. For any $n\times n$ matrix $M$, we have $I_{n}M=MI_{n}=M$, and the identity matrix is uniquely defined by this property. In addition, for any $n\times m$ matrix $A$ and $m\times n$ $B$, we have $IA=A$ and $BI=B$.

The $n\times n$ identity matrix $I$ satisfy the following properties

• For the determinant, we have $\det I=1$, and for the trace, we have $\operatorname{tr}I=n$.

• The identity matrix has only one eigenvalue $\lambda=1$ of multiplicity $n$. The corresponding eigenvectors can be chosen to be $v_{1}=(1,0,\ldots,0),\ldots,v_{n}=(0,\ldots,0,1)$.

• The matrix exponential of $I$ gives $e^{I}=eI$.

• The identity matrix is a diagonal matrix.

Title identity matrix IdentityMatrix 2013-03-22 12:06:29 2013-03-22 12:06:29 mathcam (2727) mathcam (2727) 13 mathcam (2727) Definition msc 15-01 msc 15A57 KroneckerDelta ZeroMatrix IdentityMap