Matrix of ones

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

${\displaystyle J_{2}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},\quad J_{3}={\begin{bmatrix}1&1&1\\1&1&1\\1&1&1\end{bmatrix}},\quad J_{2,5}={\begin{bmatrix}1&1&1&1&1\\1&1&1&1&1\end{bmatrix}},\quad J_{1,2}={\begin{bmatrix}1&1\end{bmatrix}}.\quad }$

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.
• The characteristic polynomial of J is ${\displaystyle (x-n)x^{n-1}}$.
• The minimal polynomial of J is ${\displaystyle x^{2}-nx}$.
• The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity n − 1. ^[4]
• ${\displaystyle J^{k}=n^{k-1}J}$ for ${\displaystyle k=1,2,\ldots .}$ ^[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 ${\displaystyle {\tfrac {1}{n}}J}$ is idempotent. ^[5]
• The matrix exponential of J is ${\displaystyle \exp(\mu J)=I+{\frac {e^{\mu n}-1}{n}}J}$

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.

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 ${\displaystyle (a\cdot b)\cdot (b\cdot c)=b}$. Finite central groupoids have a square number of elements, and the corresponding logical matrices exist only for those dimensions. ^[8]

See also

• Zero matrix, a matrix where all entries are zero
• Single-entry matrix

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. 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 (4): 376–390, doi:10.1016/S0021-9800(70)80032-1, MR   0259000