Matrix of ones

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In mathematics, a matrix of ones or all-ones matrix is a matrix over the real numbers where every element is equal to one.[1] Examples of standard notation are given below:

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[edit]

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

  • The characteristic polynomial of J is .
  • 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 with multiplicity 1 and 0 with multiplicity n−1.[4]
  • J is positive semi-definite matrix. This follows from the previous property.
  • [5]
  • The matrix is idempotent. This is a simple corollary of the above.[5]
  • 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[edit]

  1. ^ Horn, Roger A.; Johnson, Charles R. (2012), "0.2.8 The all-ones matrix and vector", Matrix Analysis, Cambridge University Press, p. 8, ISBN 9780521839402 .
  2. ^ Weisstein, Eric W. "Unit Matrix". MathWorld. 
  3. ^ Stanley, Richard P. (2013), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, Lemma 1.4, p. 4, ISBN 9781461469988 .
  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, ISBN 9780387227719 .
  6. ^ Smith, Jonathan D. H. (2011), Introduction to Abstract Algebra, CRC Press, p. 77, ISBN 9781420063721 .
  7. ^ Godsil, Chris (1993), Algebraic Combinatorics, CRC Press, Lemma 4.1, p. 25, ISBN 9780412041310 .