Difference between revisions of "Identity matrix"

Latest revision as of 23:00, 16 March 2010

In linear algebra, the square identity matrix is a $n \times n$ matrix with $1$ s in its main diagonal and $0$ s in every other entry. It is usually denoted $I_n$ .

$I_n = \begin{pmatrix}1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\end{pmatrix}.$

The corresponding linear map is the identity map. For any $n \times n$ matrix $A$ , we have $AI_n = I_nA = A$ . The inverse $A^{-1}$ of $A$ is the unique matrix such that $AA^{-1} = A^{-1}A = I_n$ .

The determinant of $I_n$ is $1$ . $I_n$ has only one eigenvalue $1$ , occurring with multiplicity $n$ . Hence, any $n \times n$ matrix is in the corresponding eigenspace. The characteristic polynomial of $I_n$ is $P_{I}(t) = (t-1)^n$ , and the minimal polynomial is $t - 1$ .

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Difference between revisions of "Identity matrix"

Latest revision as of 23:00, 16 March 2010