Unit Matrix

In mathematics , a unit matrix is a matrix with each element equal to one. Examples:

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

{\ displaystyle 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 & 1 \ end {pmatrix}}. \ quad}

{\ displaystyle 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 & 1 \ end {pmatrix}}. \ quad}

Properties

For a square n × n matrix of units of J , the following statements are true:

The trace of the matrix J is n , and the determinant is 1 for n = 1, and 0 in all other cases.
The rank of the matrix J is 1.
The matrix J has only two eigenvalues : n (non-multiple) and 0 (multiplicities n -1).
$J^{k}=n^{k-1}J,{\mbox{ for }}k=1,2,\ldots .$ ${\ displaystyle J ^ {k} = n ^ {k-1} J, {\ mbox {for}} k = 1,2, \ ldots.}$
Matrix ${\tfrac {1}{n}}J$ ${\ displaystyle {\ tfrac {1} {n}} J}$ is idempotent .
The exponent from the matrix of units is represented as: $\exp(J)=I+{\frac {e^{n}-1}{n}}J.$ ${\ displaystyle \ exp (J) = I + {\ frac {e ^ {n} -1} {n}} J.}$
J is a unit element with respect to the Hadamard product .