identity matrixIdentityMatrix2002-01-04 18:26:422006-10-25 00:24:28Definition\usepackage{amssymb}
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\usepackage{amsfonts}
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\usepackage{xypic}The $n \times n$ \emph{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 $R$.
\subsubsection{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
\begin{itemize}
\item For the determinant, we have $\det I = 1$, and for the trace, we have
$\operatorname{tr}I = n$.
\item The identity matrix has only one eigenvalue $\lambda =1$ of
multiplicity $n$. The corresponding eigenvectors can be chosen to be
$v_1=(1,0,\ldots, 0),\ldots, v_n=(0,\ldots, 0,1)$.
\item The matrix exponential of $I$ gives $e^I = e I$.
\item The identity matrix is a diagonal matrix.
\end{itemize}