Matrix of ones

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{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

In special contexts, the term unit matrix is used as a synonym for "matrix of ones"[1] 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 1 if n is 1, or 0 otherwise.
  • The rank of U is 1 and the eigenvalues are n (once) and 0 (n-1 times).
  • U^k = n^{k-1} U, \mbox{ for } k=1,2,\ldots.\,
  • The matrix \tfrac1n 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.
  • Multiplication by U with the Hadamard product is the identity operator.

References

  1. ^ Weisstein, Eric W., "Unit Matrix" from MathWorld.
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