Unit matrix

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In mathematics, the unit matrix is a matrix where every element is equal to one. Examples of standard notation are given below:

$J_2=\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix};\quad J_3=\begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix};\quad J_{2,5}=\begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{pmatrix}.\quad$

Note that a minority of authors use the term unit matrix to denote the identity matrix. While there is a certain appeal in this practice, especially when viewed in the context of abstract algebra where the idea of a unit is analogous to the identity matrix, such use is generally undesirable as no other commonly understood term for the unit matrix exists, and the term identity matrix has been almost universally adopted.

[edit] Properties

For an n×n unit matrix U, the following conditions hold:

$U^k = n^{k-1} U, \mbox{ for } k=1,2,\ldots.\,$
The matrix ${1\over n}U$ is idempotent. This is a simple corollary of the above.
$\operatorname{exp}(U) = I + \frac{ e^n-1}{n} U,$ where exp(U) is the matrix exponential.