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

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For the determinant, we have $\det I=1$ , and for the trace, we have $\operatorname{tr}I=n$ .
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The identity matrix has only one eigenvalue $\lambda=1$ of multiplicity $n$ . The corresponding eigenvectors can be chosen to be $v_{1}=(1,0,\ldots,0),\ldots,v_{n}=(0,\ldots,0,1)$ .
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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	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