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

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

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

• The identity matrix has only one eigenvalue $\lambda =1$ ofmultiplicity $n$. The corresponding eigenvectors can be chosen to be$v_1=(1,0,\mathrm{\dots },0),\mathrm{\dots },v_n=(0,\mathrm{\dots },0,1)$.

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

• The identity matrix is a diagonal matrix.