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identity matrix

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

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

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

The identity matrix

serves as the multiplicative identity in the ring of $n\times n$ matrices over

with standard matrix multiplication. For any $n\times n$ matrix

, we have

, and the identity matrix is uniquely defined by this property. In addition, for any $n\times m$ matrix

and $m\times n$

, we have

and

The $n\times n$ identity matrix 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 . 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 gives .
The identity matrix is a diagonal matrix.

"identity matrix" is owned by mathcam. [ full author list (2) | owner history (1) ]

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AMS MSC:	15A57 (Linear and multilinear algebra; matrix theory :: Other types of matrices )
	15-01 (Linear and multilinear algebra; matrix theory :: Instructional exposition )

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