Inmathematics, amatrix of ones orall-ones matrix is amatrix with every entry equal toone.^[1] For example:

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

Avector of ones orall-ones vector is matrix of ones havingrow or column form; it should not be confused withunit vectors.

For ann ×n matrix of onesJ, the following properties hold:

• The trace ofJ equalsn,^[3] and thedeterminant equals 0 forn ≥ 2, but equals 1 ifn = 1.
• Thecharacteristic polynomial ofJ is${\displaystyle (x-n)x^{n-1}}$.
• Theminimal polynomial ofJ is${\displaystyle x^2-nx}$.
• Therank ofJ is 1 and theeigenvalues aren withmultiplicity 1 and 0 with multiplicityn − 1.^[4]
• ${\displaystyle J^k=n^{k-1}J}$ for${\displaystyle k=1,2,\dots .}$^[5]
• J is theneutral element of theHadamard product.^[6]

WhenJ is considered as a matrix over thereal numbers, the following additional properties hold:

• J ispositive semi-definite matrix.
• The matrix${\displaystyle \frac{1}{n}J}$ isidempotent.^[5]
• Thematrix exponential ofJ is${\displaystyle \exp (\mu J)=I+\frac{e^{\mu n}-1}{n}J}$

The all-ones matrix arises in the mathematical field ofcombinatorics, particularly involving the application of algebraic methods tograph theory. For example, ifA is theadjacency matrix of ann-vertexundirected graphG, andJ is the all-ones matrix of the same dimension, thenG is aregular graph if and only ifAJ = JA.^[7] As a second example, the matrix appears in some linear-algebraic proofs ofCayley's formula, which gives the number ofspanning trees of acomplete graph, using thematrix tree theorem.

The logicalsquare roots of a matrix of ones,logical matrices whose square is a matrix of ones, can be used to characterize thecentral groupoids. Central groupoids are algebraic structures that obey theidentity${\displaystyle (a\cdot b)\cdot (b\cdot c)=b}$. Finite central groupoids have asquare number of elements, and the corresponding logical matrices exist only for those dimensions.^[8]

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

• Matrices (mathematics)
• 1 (number)

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