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
$$I=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\ddots}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 1\hfill \end{array}\right],$$ 
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 ${I}_{n}M=M{I}_{n}=M$, 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

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For the determinant^{}, we have $detI=1$, and for the trace, we have $\mathrm{tr}I=n$.

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The identity matrix has only one eigenvalue^{} $\lambda =1$ of multiplicity $n$. The corresponding eigenvectors^{} can be chosen to be ${v}_{1}=(1,0,\mathrm{\dots},0),\mathrm{\dots},{v}_{n}=(0,\mathrm{\dots},0,1)$.

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

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The identity matrix is a diagonal matrix^{}.
Title  identity matrix 

Canonical name  IdentityMatrix 
Date of creation  20130322 12:06:29 
Last modified on  20130322 12:06:29 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  13 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 1501 
Classification  msc 15A57 
Related topic  KroneckerDelta 
Related topic  ZeroMatrix 
Related topic  IdentityMap 