Matrix of ones

Matrix where every entry is equal to one

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

J 2 = ( 1 1 1 1 ) ; J 3 = ( 1 1 1 1 1 1 1 1 1 ) ; J 2 , 5 = ( 1 1 1 1 1 1 1 1 1 1 ) ; J 1 , 2 = ( 1 1 ) . {\displaystyle 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 J_{1,2}={\begin{pmatrix}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 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

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

  • The trace of J equals n,[3] and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1.[a]
  • The characteristic polynomial of J is ( x n ) x n 1 {\displaystyle (x-n)x^{n-1}} .
  • The minimal polynomial of J is x 2 n x {\displaystyle x^{2}-nx} .
  • The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1.[4]
  • J k = n k 1 J {\displaystyle J^{k}=n^{k-1}J} for k = 1 , 2 , . {\displaystyle k=1,2,\ldots .} [5]
  • J is the neutral element of the Hadamard product.[6]

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

  • J is positive semi-definite matrix.
  • The matrix 1 n J {\displaystyle {\tfrac {1}{n}}J} is idempotent.[5]
  • The matrix exponential of J is exp ( J ) = I + e n 1 n J . {\displaystyle \exp(J)=I+{\frac {e^{n}-1}{n}}J.}

Applications

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.

See also

References

  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.
  1. ^ One may also consider the case n = 0, in which case the empty matrix is vacuously an all-ones matrix, also with determinant 1.[citation needed]
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