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Matrix of ones

In mathematics, the matrix of ones, also known as the all-ones matrix and commonly denoted by J, is a matrix in which every entry is equal to 1.^[1] For an m \times n matrix, it takes the form J_{m \times n} with all elements 1, and the square case J_n (where m = n) is particularly prominent in applications.^[2] This matrix is symmetric when square and is closely related to the adjacency matrix of the complete graph K_n, which is J_n - I_n where I_n is the n \times n identity matrix.^[3] Key algebraic properties of the all-ones matrix include its low rank and simple spectral structure. The rank of J_n is 1, as all rows (or columns) are identical and thus linearly dependent, spanning a one-dimensional space generated by the all-ones vector.^[1]^[2] Its eigenvalues consist of n with algebraic multiplicity 1 (corresponding to the eigenvector of all ones) and 0 with multiplicity n-1, reflecting its structure as an outer product of the all-ones vectors: J_n = \mathbf{1}_n \mathbf{1}_n^T, where \mathbf{1}_n is the n \times 1 vector of ones.^[2] Additionally, J_n^2 = n J_n, and normalizing by n yields a doubly stochastic matrix whose rows and columns each sum to 1, useful in probability models.^[1] The matrix of ones appears in diverse applications across mathematics and related fields. In linear programming and optimization, it arises in slack variable formulations and dual problems, facilitating constraint handling.^[2] In statistics and least-squares regression, it models constant terms, with residuals orthogonal to J ensuring their sum is zero.^[2] For Markov chains, the normalized J_n / n represents a transition matrix for uniform stationary distributions, aiding steady-state analysis.^[1] In numerical methods, such as finite differences for multidimensional problems like Laplace's equation, it features in Kronecker products for grid-based computations.^[2]

Introduction

Definition

In linear algebra, the matrix of ones, also known as the all-ones matrix, is an m \times n matrix J_{m,n} where every entry (i,j) equals 1, for positive integers m and n.^[4] This matrix is rectangular in general, reducing to a square n \times n matrix when m = n. For example, the $2 \times 2 matrix of ones is

J_2 = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix},

while the $2 \times 3 rectangular case is

J_{2,3} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}.

^[1] In general, J_{m,n} can be expressed as the outer product of the all-ones column vector of length m and the all-ones row vector of length n, denoted J_{m,n} = \mathbf{1}_{m \times 1} \mathbf{1}_{1 \times n}^T, where \mathbf{1}_k is the vector of k ones.^[5] The matrix of ones, denoted J_n, is distinct from the identity matrix I_n, which has 1s only on the main diagonal and 0s elsewhere.^[6] This structural difference results in fundamentally different multiplication behaviors; for instance, multiplying any matrix by I_n leaves it unchanged, whereas J_n scales and sums components across all entries.^[7] While the all-ones vector, often denoted \mathbf{1}_n, is a column vector (or row vector) with every entry equal to 1, the matrix of ones J_{n,n} is the two-dimensional outer product \mathbf{1}_n \mathbf{1}_n^T.^[1] This relation highlights how the matrix extends the vector concept to a full rank-one structure, but the vector itself is a one-dimensional object used in contexts like inner products or basis expansions.^[1] The term "unit matrix" can cause confusion, as some older texts apply it to the matrix of ones, but in standard linear algebra, it refers to the identity matrix.^[8] For example, Akivis and Goldberg (1972) use "unit matrix" synonymously with the identity, underscoring the need to clarify context in historical or non-standard sources.^[7] In contrast to the matrix of ones, the zero matrix has all entries equal to 0, serving as its additive inverse and the trivial counterpart in matrix spaces.^[1]

Properties

Algebraic Properties

The matrix of ones J_{m,n}, an m \times n matrix with all entries equal to 1, has rank 1 whenever m \geq 1 and n \geq 1, as it admits the factorization J_{m,n} = \mathbf{1}_m \mathbf{1}_n^T, the outer product of the all-ones column vectors of lengths m and n.^[9] For the square case J_n, this implies a nullity of n-1.^[9] The trace of the square matrix J_n is n, equal to the sum of its n diagonal entries, each of which is 1.^[9] Under matrix multiplication, the square matrix satisfies J_n^2 = n J_n.^[9] More generally, for compatible rectangular matrices, J_{p,q} J_{q,r} = q J_{p,r}.^[9] Scalar multiplication by a constant c yields the matrix with all entries equal to c. Addition of the identity matrix gives J_n + I_n, which has 1s on the main diagonal and 2s in all off-diagonal positions.^[9] The square matrix J_n is singular and thus not invertible for all n > 1, owing to its rank of 1 being strictly less than n.^[9] Its Moore-Penrose pseudoinverse is J_n^+ = \frac{1}{n^2} J_n.^[10]

Spectral Properties

The all-ones matrix J_n, an n \times n matrix with every entry equal to 1, has a simple spectrum consisting of a single eigenvalue n with algebraic multiplicity 1 and the eigenvalue 0 with algebraic multiplicity n-1.^[11] This structure arises from the rank-1 nature of J_n, which implies at most one nonzero eigenvalue.^[11] The eigenvector corresponding to the eigenvalue n is the all-ones vector \mathbf{e}_n = [1, 1, \dots, 1]^T, since J_n \mathbf{e}_n = n \mathbf{e}_n.^[11] The eigenspace for the eigenvalue 0 is the orthogonal complement of \operatorname{span}\{\mathbf{e}_n\}, consisting of all vectors \mathbf{x} \in \mathbb{R}^n such that \sum_{i=1}^n x_i = 0, or equivalently, the kernel of J_n, which has dimension n-1.^[11] The characteristic polynomial of J_n is p_{J_n}(\lambda) = \det(\lambda I_n - J_n) = \lambda^{n-1} (\lambda - n).^[11] To derive this, note that J_n = \mathbf{e}_n \mathbf{e}_n^T, so \lambda I_n - J_n = \lambda I_n - \mathbf{e}_n \mathbf{e}_n^T; by the matrix determinant lemma, \det(\lambda I_n - \mathbf{e}_n \mathbf{e}_n^T) = \lambda^n (1 - \mathbf{e}_n^T (\lambda^{-1} I_n) \mathbf{e}_n ) = \lambda^n (1 - n/\lambda) = \lambda^{n-1} (\lambda - n) for \lambda \neq 0, and the case \lambda = 0 follows by continuity or direct computation.^[11] Since J_n possesses a full set of n linearly independent eigenvectors—one for \lambda = n and n-1 for \lambda = 0—it is diagonalizable over the reals.^[11] Specifically, there exists an invertible matrix P whose columns are these eigenvectors such that J_n = P D P^{-1}, where D = \operatorname{diag}(n, 0, \dots, 0).^[11] The determinant of J_n is \det(J_n) = 0 for all n > 1, reflecting its singularity due to the eigenvalue 0 having multiplicity greater than 0; for n = 1, \det(J_1) = 1.^[11]

Representations and Constructions

As Rank-One Matrices

The all-ones matrix J_{m,n}, an m \times n matrix with every entry equal to 1, can be expressed as the outer product of an m \times 1 all-ones vector \mathbf{1}_m and an n \times 1 all-ones vector \mathbf{1}_n, specifically J_{m,n} = \mathbf{1}_m \mathbf{1}_n^T. This form arises because each entry (J_{m,n})_{ij} = 1 = (\mathbf{1}_m)_i \cdot (\mathbf{1}_n)_j, filling the matrix uniformly with ones. In general, any rank-one matrix can be represented as the outer product \mathbf{u} \mathbf{v}^T for non-zero column vectors \mathbf{u} and \mathbf{v}; the all-ones matrix is the particular case where both \mathbf{u} = \mathbf{1}_m and \mathbf{v} = \mathbf{1}_n, ensuring all entries are identical and positive. This representation directly implies that J_{m,n} has rank one, as the column space is one-dimensional, spanned solely by \mathbf{1}_m. For rectangular matrices where m \neq n, the rank remains one, with the column space spanned by \mathbf{1}_m and the row space by \mathbf{1}_n. In numerical software such as MATLAB or NumPy, J_{m,n} is often generated efficiently via outer product computations or array broadcasting, avoiding explicit entry-wise assignments for large dimensions.

Relations to Other Special Matrices

The all-ones matrix J_n can be expressed as the sum of the identity matrix I_n and the matrix J_n - I_n, where the latter serves as the adjacency matrix of the complete graph K_n on n vertices.^[12] This decomposition highlights the structural connection between the uniform connectivity of the complete graph and the all-ones structure, with diagonal elements adjusted to reflect the absence of self-loops. The sum of all n! permutation matrices of order n equals (n-1)! J_n, as each entry (i,j) receives a contribution from exactly (n-1)! permutations that map row i to column j.^[13] This relation underscores the all-ones matrix as a scaled aggregate of permutation matrices, linking it to the symmetric group S_n. In statistics, the centering matrix H_n = I_n - \frac{1}{n} J_n projects vectors onto the orthogonal complement of the all-ones vector \mathbf{1}_n, effectively subtracting the mean from each component.^[14] This idempotent operator H_n (with H_n^2 = H_n) is fundamental for mean-centering data matrices, isolating deviations from the constant subspace spanned by \mathbf{1}_n. The matrix \frac{1}{n} J_n is idempotent, satisfying \left( \frac{1}{n} J_n \right)^2 = \frac{1}{n} J_n, since J_n^2 = n J_n. This property positions \frac{1}{n} J_n as a rank-one projection onto the span of \mathbf{1}_n, contrasting with the zero matrix O_n, which is the trivial idempotent of rank zero. Among constant-entry square matrices over \{0,1\}, J_n achieves the maximum possible sum of entries (n^2), serving as the extremal case opposite to the zero matrix O_n with minimal sum (0).

Applications

In Linear Algebra and Statistics

The matrix of ones, denoted J_n, plays a fundamental role in linear algebra by facilitating projections and summations. For any vector \mathbf{x} \in \mathbb{R}^n, multiplication by J_n yields J_n \mathbf{x} = \left( \sum_{i=1}^n x_i \right) \mathbf{e}_n, where \mathbf{e}_n is the all-ones vector in \mathbb{R}^n. This operation effectively scales the all-ones vector by the sum of the components of \mathbf{x}, highlighting J_n's utility in aggregating information across dimensions.^[15] A key application arises in orthogonal projections, where the matrix \frac{1}{n} J_n serves as the projection operator onto the one-dimensional subspace spanned by \mathbf{e}_n, known as the all-ones subspace. This projector is symmetric and idempotent, with eigenvalues 1 (multiplicity 1) and 0 (multiplicity n-1), confirming its status as an orthogonal projection. In least squares estimation, \frac{1}{n} J_n projects data onto constant models, estimating the overall mean by minimizing the squared error to a constant vector, which is essential for baseline fitting in regression contexts.^[16] In statistics, the centering matrix I_n - \frac{1}{n} J_n is used to demean data by subtracting row or column means, producing mean-zero vectors or matrices that are orthogonal to \mathbf{e}_n. This operation, (I_n - \frac{1}{n} J_n) \mathbf{x} = \mathbf{x} - \bar{x} \mathbf{e}_n where \bar{x} = \frac{1}{n} \sum x_i, ensures centered data for unbiased covariance estimation and is idempotent with rank n-1. In analysis of variance (ANOVA) and linear regression, J_n appears implicitly through the design matrix, which includes a column of all ones to model the intercept term, representing the baseline mean across observations. For instance, in the overall mean model of ANOVA, the design matrix reduces to a column vector of ones, enabling estimation of group means via generalized least squares.^[16]^[17]^[18] In principal component analysis (PCA), J_n facilitates data centering, where the centered data matrix is (I_n - \frac{1}{n} J_n) X for an n \times p observation matrix X, removing the mean from each feature. This step aligns with the covariance matrix computation, as uncentered data incorporate a rank-one term from the outer product of the all-ones vector and the mean vector \boldsymbol{\mu}, namely \mathbf{e}_n \boldsymbol{\mu}^T, which J_n helps isolate before subtraction to focus on variance structure.^[19]

In Graph Theory and Combinatorics

In graph theory, the matrix of ones, denoted J_n, plays a fundamental role in representing the complete graph K_n. The adjacency matrix A of K_n is given by A = J_n - I_n, where I_n is the n \times n identity matrix, since every pair of distinct vertices is connected by an edge, with no self-loops.^[20] This structure highlights J_n's utility in encoding fully connected discrete structures. For regular graphs, the matrix of ones relates to the adjacency matrix through eigenvector properties. A graph is d-regular if and only if its adjacency matrix A satisfies A \mathbf{j} = d \mathbf{j}, where \mathbf{j} is the all-ones vector, making \mathbf{j} an eigenvector corresponding to the largest eigenvalue d. Equivalently, A J_n = d J_n, reflecting the commuting property and uniformity of degrees.^[21] This connection is central in spectral graph theory for analyzing regularity and connectivity. The matrix of ones appears in the Laplacian matrix of the complete graph, L = n I_n - J_n, where the off-diagonal entries are -1 and diagonals are n-1. By Kirchhoff's matrix-tree theorem, the number of spanning trees in a graph equals any cofactor of its Laplacian. For K_n, all cofactors equal n^{n-2}, yielding Cayley's formula for the number of labeled trees on n vertices.^[22]^[23] In enumerative combinatorics, the permanent of J_n is n!, counting all permutations as each term in the sum is 1. More relevant to counting problems, the permanent of J_n - I_n equals the number of derangements !n, the permutations with no fixed points, connecting to inclusion-exclusion principles where terms like \sum (-1)^k \binom{n}{k} (n-k)! approximate derangements asymptotically as n!/e. The all-ones structure facilitates such counts in permutation avoidance and rook placements on chessboards without diagonal attacks.^[24] Logical square roots of the matrix of ones, (0,1)-matrices B such that B^2 = J_n, characterize central groupoids, algebraic structures where every pair of elements has a unique product equaling a central element. These matrices correspond to central digraphs, directed graphs with a unique length-two path between any ordered pair of distinct vertices, linking combinatorial designs to idempotent binary operations with square order n.^[25]

References

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[PDF] Discover Linear Algebra - Full-Time Faculty
Jan 15, 2023 · Notation 2.1.7 (All-ones matrix). The k × n matrix with all of its entries equal to 1 is denoted by. Jk×n or J. We write Jn for Jn×n ...
[2]
[PDF] Linear Algebra and It's Applications by Gilbert Strang
... all-ones matrix A = ones(4,4) and the column v = ones(4,1), what is A ∗ v? (Computer not needed.) If you multiply B = eye(4). + ones(4,4) times w = zeros(4 ...
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[PDF] Matrix Analysis, Pólya frequency functions, and Preservers of (Total ...
... Matrix Analysis and Positivity, a ... Three examples of real matrices are 0m×n,1m×n,Idn×n, which are the (rectangular) matrix consisting of all zeros, all ones, ...
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Identity Matrix -- from Wolfram MathWorld
The identity matrix is a the simplest nontrivial diagonal matrix, defined such that I(X)=X (1) for all vectors X. An identity matrix may be denoted 1, I, ...
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Unit Matrix -- from Wolfram MathWorld
### Summary of Unit Matrix from Wolfram MathWorld
[6]
None
Below is a merged summary of the information on the "all-ones matrix" (J = ee^T) from *Matrix Analysis* by Roger A. Horn (2nd Edition) and related sources, consolidating all details from the provided segments. To maximize density and clarity, I will use tables in CSV format where appropriate, followed by a narrative summary for additional context. The response retains all mentioned information, cross-referencing page numbers, properties, and URLs as provided.
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[PDF] The Moore-Penrose Pseudoinverse (Math 33A: Laub)
In these notes we give a brief introduction to the Moore-Penrose pseudoinverse, a gen- eralization of the inverse of a matrix. The Moore-Penrose ...
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https://people.tamu.edu/~yvorobets//MATH323-2022C/Lect1-04web.pdf
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[PDF] arXiv:1202.4525v2 [math.FA] 17 Apr 2012
Apr 17, 2012 · JM+1, where JM+1 denotes an (M + 1) × (M + 1) matrix of ones. Notice that the spectrum of Ψ∗Ψ consists of M copies of M+1. M and one value ...
[10]
Extensions of biclique decomposition
The adjacency matrix of the complete graph $K_n$ is $J_n-I_n$, where $J_n$ is the $n$-by-$n$ matrix with every entry $1$. Its eigenvalues are $n-1$ with ...<|separator|>
[11]
Sum of Traces of all Permutations of a Matrix - Math Stack Exchange
Apr 26, 2020 · First of all, I claim that the sum of all order n permutations matrices is equal to the matrix whose entries are all equal to (n−1)!
[12]
[PDF] An eigenanalysis of data centering in machine learning - arXiv
The matrix I is the n-by-n identity matrix, and the vector. 1 is the all-ones vector of n entries. Then, 11⊤ is the n- by-n matrix of all ones, and 1⊤1 = n.
[13]
[PDF] Introduction to Matrix-based Data Science: Mathematics, Computing ...
Jan 18, 2022 · The maximal number of linearly independent ... = a b c 1 − a . It should also be noted that the constant matrix of 1's, Jn, is not idempotent:.
[14]
3.3 Representing Summation Using Matrix Notation - Bookdown
In R, summing the elements in a vector can be done using matrix algebra. # create vector of 1's and a vector of x onevec = rep(1,3) onevec. ## [1] 1 1 1. xvec ...Missing: Jx = | Show results with:Jx =<|control11|><|separator|>
[15]
[PDF] Matrix Algebra - San Jose State University
Any nonzero row or column vector (as a matrix) has rank 1. • A nonzero matrix is of rank 1 if and only if all of its nonzero rows. (or columns) are multiples of ...
[16]
4: ANOVA Models Part II - STAT ONLINE
In the GLM model, Y = X β + E , the design matrix X for the overall mean model turns out to be a 6-dimensional column vector of ones. The parameter vector, β is ...
[17]
Lecture 12: Matrix Notation for Regression
1. Model and notation · 2. Matrix operations · 3. Estimating the mean · 4. Estimating the variance · 5. Adding a column.
[18]
Principal Component Analysis (PCA) & Autoenocoders - MATH-CS ...
... C N = I N − 1 N 1 N 1 N ⊤ is the centering matrix. Note: J denotes the N × N all-ones matrix: J = 1 N 1 N ⊤ . Note: To verify the double centering ...
[19]
[PDF] Minimum Roman Dominating Distance Energy of a Graph - arXiv
Mar 31, 2022 · The adjacency matrix of a complete graph Kn is Jn −In and the eigenvalues are n − 1 and −1 with multiplicity n − 1, the energy of Kn is.Missing: J_n - | Show results with:J_n -
[20]
[PDF] ON THE PRINCIPAL EIGENVECTOR OF A GRAPH YUEHENG ...
Dec 31, 2019 · Recall that j denotes the all-ones vector. Corollary 2.22. j is an eigenvector of A if and only if G is regular. Proof. Suppose G is d-regular.
[21]
[PDF] Lecture 19 1 Matrix-Tree Theorem
Apr 21, 2011 · Definition 1 (Laplacian matrix of undirected graph) The Laplacian matrix L of G ....... = nI - J, where J is the n × n matrix of ones. J ...
[22]
[PDF] Matrix Tree Theorem and more - MIT OpenCourseWare
By the way, we can get Cayley's formula from the matrix tree theorem! Remember that if G = Kn, the complete graph on n vertices, the Laplacian matrix has all ...
[23]
Permanent -- from Wolfram MathWorld
The permanent of a matrix A is the coefficient of x_1...x_n in. product_(i=1)^n ... n×n (0,1)-matrix is n! , corresponding to all elements 1. See also.
[24]
[PDF] Central Groupoids, Central Digraphs - Chi-Kwong Li
x — matrix of all ones. These two concepts are also ...