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Matrix of ones

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Title: Matrix of ones  
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Matrix of ones

In mathematics, a matrix of ones or all-ones matrix is a matrix where every element is equal to one.[1] 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

Some sources call the all-ones matrix the unit matrix,[2] but that term may also refer to the identity matrix, a different matrix.

Properties

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

  • The trace of J is n,[3] and the determinant is 1 if n is 1, or 0 otherwise.
  • The rank of J is 1 and the eigenvalues are n (once) and 0 (n-1 times).[4]
  • J is positive semi-definite matrix. This follows from the previous property.
  • J^k = n^{k-1} J, \mbox{ for } k=1,2,\ldots.\,[5]
  • The matrix \tfrac1n J is idempotent. This is a simple corollary of the above.[5]
  • \exp(J) = I + \frac{ e^n-1}{n} J, where exp(J) is the matrix exponential.
  • J is the neutral element of the Hadamard product.[6]
  • If A is the adjacency matrix of a n-vertex undirected graph G, and J is the all-ones matrix of the same dimension, then G is a regular graph if and only if AJ = JA.[7]

References

  1. ^ Horn, Roger A.;  .
  2. ^ Weisstein, Eric W., "Unit Matrix", MathWorld.
  3. ^  .
  4. ^ Stanley (2013); Horn & Johnson (2012), p. 65.
  5. ^ a b Timm, Neil H. (2002), Applied Multivariate Analysis, Springer texts in statistics, Springer, p. 30,  .
  6. ^ Smith, Jonathan D. H. (2011), Introduction to Abstract Algebra, CRC Press, p. 77,  .
  7. ^  .
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