In mathematics, a matrix of ones or all-ones matrix is a matrix with every entry equal to one.. For example:
Some sources call the all-ones matrix the unit matrix, 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 matrix of ones J, the following properties hold:
The trace of J equals n,. and the determinant equals 0 for n ≥ 2, but equals 1 if n = 1. The characteristic polynomial of J is . The minimal polynomial of J is . The rank of J is 1 and the eigenvalues are n with multiplicity 1 and 0 with multiplicity .; , p. 65. for . J is the neutral element of the Hadamard product..
When J is considered as a matrix over the real numbers, the following additional properties hold: J is positive semi-definite matrix. The matrix is idempotent. The matrix exponential of J is
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.. 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.
See also Zero matrix, a matrix where all entries are zero Single-entry matrix
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