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 $R$ .

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 $I_nM=MI_n=M$ , 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 $e^I = e I$ .
• The identity matrix is a diagonal matrix.