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Matrix of ones

From Wikipedia, the free encyclopedia

In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one.[1] For example:

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

A vector of ones or all-ones vector is matrix of ones having row or column form; it should not be confused with unit vectors.

Properties

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

When J is considered as a matrix over the real numbers, the following additional properties hold:

Applications

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The all-ones matrix arises in the mathematical field of combinatorics, particularly involving the application of algebraic methods to graph theory. For example, if A is the adjacency matrix of an 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] As a second example, the matrix appears in some linear-algebraic proofs of Cayley's formula, which gives the number of spanning trees of a complete graph, using the matrix tree theorem.

The logical square roots of a matrix of ones, logical matrices whose square is a matrix of ones, can be used to characterize the central groupoids. Central groupoids are algebraic structures that obey the identity . Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions.[8]

See also

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References

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  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.
  8. ^ Knuth, Donald E. (1970), "Notes on central groupoids", Journal of Combinatorial Theory, 8: 376–390, doi:10.1016/S0021-9800(70)80032-1, MR 0259000