Matrix of ones
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:
- [math]\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 }[/math]
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.[lower-alpha 1]
- The characteristic polynomial of J is [math]\displaystyle{ (x - n)x^{n-1} }[/math].
- The minimal polynomial of J is [math]\displaystyle{ x^2-nx }[/math].
- The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1.[4]
- [math]\displaystyle{ J^k = n^{k-1} J }[/math] for [math]\displaystyle{ k = 1,2,\ldots . }[/math][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 [math]\displaystyle{ \tfrac1n J }[/math] is idempotent.[5]
- The matrix exponential of J is [math]\displaystyle{ \exp(J) = I + \frac{ e^n-1}{n} J. }[/math]
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
- Zero matrix, a matrix where all entries are zero
- Single-entry matrix
References
- ↑ 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, https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA8.
- ↑ Weisstein, Eric W.. "Unit Matrix". http://mathworld.wolfram.com/UnitMatrix.html.
- ↑ Stanley, Richard P. (2013), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, Lemma 1.4, p. 4, ISBN 9781461469988, https://books.google.com/books?id=_Tc_AAAAQBAJ&pg=PA4.
- ↑ (Stanley 2013); (Horn Johnson), p. 65.
- ↑ 5.0 5.1 Timm, Neil H. (2002), Applied Multivariate Analysis, Springer texts in statistics, Springer, p. 30, ISBN 9780387227719, https://books.google.com/books?id=vtiyg6fnnskC&pg=PA30.
- ↑ Smith, Jonathan D. H. (2011), Introduction to Abstract Algebra, CRC Press, p. 77, ISBN 9781420063721, https://books.google.com/books?id=PQUAQh04lrUC&pg=PA77.
- ↑ Godsil, Chris (1993), Algebraic Combinatorics, CRC Press, Lemma 4.1, p. 25, ISBN 9780412041310, https://books.google.com/books?id=eADtlNCkkIMC&pg=PA25.
- ↑ 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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