Definitions

# Matrix of ones

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

$J_2=begin\left\{pmatrix\right\}$
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

In special contexts, the term unit matrix is used as a synonym for "matrix of ones This is done whenever it is clear that "unit matrix" does not refer to the identity matrix.

## Properties

For an n×n matrix of ones U, the following properties hold:

• The trace of U is n, and the determinant is zero.
• The rank of U is 1 and the eigenvalues are n (once) and 0 (n-1 times).
• $U^k = n^\left\{k-1\right\} U, mbox\left\{ for \right\} k=1,2,ldots.,$
• The matrix $tfrac1n U$ is idempotent. This is a simple corollary of the above.
• $operatorname\left\{exp\right\}\left(U\right) = I + frac\left\{ e^n-1\right\}\left\{n\right\} U,$ where exp(U) is the matrix exponential.
• Multiplication by U with the Hadamard product is the identity operator.

## References

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