Rank-1 Matrix Permanents

• Rank-1 matrix permanents are defined for matrices of the form uvᵀ, where the permanent simplifies to n! times the product of the entries.
• They underpin sharp extremal bounds in (±1)-matrix theory and yield closed-form expressions for structured perturbations like sI+tJ.
• Their exact factorization bridges rigorous permanent evaluation with efficient approximation methods such as Bethe and Sinkhorn permanents in low nonnegative rank settings.

Searching arXiv for recent and foundational papers on rank-1 matrix permanents and related low-rank permanent results. Rank-1 matrix permanents concern the permanent of matrices of the form $A=u v^{\top}$, where the matrix has linear rank $1$. In this regime, the permutation sum defining the permanent collapses to a single monomial multiplied by $n!$, because every diagonal product is identical. This elementary factorization is exact over arbitrary fields for the algebraic rank-1 identity, and it becomes the basis for several distinct research threads: sharp extremal bounds for $(\pm 1)$-matrices, bounded-rank conjectures for nonnegative and stochastic matrices, closed forms for structured perturbations such as $sI+tJ$, and approximation guarantees for Bethe and Sinkhorn permanents in low non-negative rank (Feinsilver et al., 2017, Budrevich et al., 2018, Lavi, 2018, Anari et al., 2020).

1. Exact formula and structural characterization

For an $n\times n$ matrix $A=(a_{ij})$, the permanent is

$$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$

If $A=u v^{\top}$ with $u,v\in \mathbb{F}^n$, then

$1$0

which is independent of $1$1. Summing over the $1$2 permutations yields

$1$3

The same identity implies that every diagonal product of a rank-1 matrix is equal, and the permanent is exactly $1$4 times any one of them (Feinsilver et al., 2017).

This formula specializes immediately to the all-ones matrix $1$5, giving $1$6, and more generally to $1$7, giving $1$8 (Feinsilver et al., 2017). In the nonnegative setting, if $1$9 with $n!$0, the same factorization remains valid: $n!$1 with the notational variant $n!$2 used in some papers (Lavi, 2018, Anari et al., 2020).

For rank-1 $n!$3-matrices, the outer-product form is especially rigid. If $n!$4 has rank $n!$5 and entries in $n!$6, then necessarily

$n!$7

Equivalently, every row is constant up to sign, and every column is constant up to sign. In that case,

$n!$8

so $n!$9 (Budrevich et al., 2018).

2. Extremal role in the theory of $(\pm 1)$0-matrices

Within the theory of permanents of $(\pm 1)$1-matrices over fields of characteristic zero, the rank-1 case is the base case of the sharp rank-dependent extremal theorem proved in the resolution of Kräuter’s conjecture (Budrevich et al., 2018). The paper defines $(\pm 1)$2 and, in the square case, $(\pm 1)$3, where $(\pm 1)$4 has $(\pm 1)$5 on the first $(\pm 1)$6 diagonal positions and $(\pm 1)$7 elsewhere. Thus $(\pm 1)$8.

Kräuter’s conjectural bound, proved in the paper, states that for $(\pm 1)$9 with $sI+tJ$0 and $sI+tJ$1,

$sI+tJ$2

with equality if and only if $sI+tJ$3 is obtained from $sI+tJ$4 by the standard transformations: row permutations, column permutations, transposition, and multiplication of any row or column by $sI+tJ$5 (Budrevich et al., 2018). In the rank-1 case, $sI+tJ$6, so the bound becomes

$sI+tJ$7

Because rank-1 $sI+tJ$8-matrices satisfy $sI+tJ$9 exactly, they are precisely the extremizers for the $n\times n$0 case (Budrevich et al., 2018).

This result refines the earlier theorem of Wang, which gave the general upper bound $n\times n$1 for $n\times n$2, with equality if and only if $n\times n$3 reduces to $n\times n$4 by standard transformations (Budrevich et al., 2018). The rank-1 formula therefore identifies the equality class explicitly: it is exactly the set of outer products $n\times n$5 with $n\times n$6.

The broader theorem also places rank $n\times n$7 at the top of a monotone hierarchy. For $n\times n$8, $n\times n$9 for all $A=(a_{ij})$0, so the sharp upper bound strictly decreases as rank increases (Budrevich et al., 2018). A unique anomaly occurs in the $A=(a_{ij})$1 nonsingular case, where $A=(a_{ij})$2, but the rank-1 case is unaffected (Budrevich et al., 2018).

A significant nuance concerns characteristic. The exact rank-1 identity

$A=(a_{ij})$3

is purely algebraic and holds over any field, whereas the global sharp inequalities and equality characterization in the $A=(a_{ij})$4 extremal theory are established over fields of characteristic zero (Budrevich et al., 2018).

3. Nonnegative, stochastic, and doubly stochastic rank-1 matrices

For nonnegative matrices, rank $A=(a_{ij})$5 is central in the bounded-rank framework developed for permanents and diagonal products (Lavi, 2018). If $A=(a_{ij})$6, then every diagonal product equals

$A=(a_{ij})$7

and hence

$A=(a_{ij})$8

This exact formula is the $A=(a_{ij})$9 instance of a broader program relating permanent maximization to rank constraints (Lavi, 2018).

Under the row-stochastic constraint, the structure simplifies further. If $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$0 is row-stochastic, then $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$1 for all $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$2, so $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$3 must be constant. Writing $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$4 and $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$5, one obtains

$$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$6

By AM–GM,

$$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$7

with equality if and only if $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$8 for all $$\operatorname{per}(A)=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$9. Therefore the maximum permanent over rank-1 row-stochastic matrices is

$A=u v^{\top}$0

attained uniquely by the uniform rank-1 stochastic matrix

$A=u v^{\top}$1

in the notation of that paper (Lavi, 2018).

For doubly stochastic rank-1 matrices, the feasible set collapses completely. If $A=u v^{\top}$2 is both row- and column-stochastic, then both $A=u v^{\top}$3 and $A=u v^{\top}$4 must be constant, so the only such matrix is

$A=u v^{\top}$5

and

$A=u v^{\top}$6

In this sense, the rank-1 doubly stochastic case simultaneously realizes the van der Waerden value and the maximal value permitted by the rank-1 constraint (Lavi, 2018).

The same paper formulates equivalent bounded-rank conjectures in stochastic and nonnegative forms. In the stochastic formulation, the conjectured maximizer for rank at most $A=u v^{\top}$7 is a block-diagonal doubly stochastic matrix of composition type; when specialized to $A=u v^{\top}$8, this reduces exactly to $A=u v^{\top}$9, with extremal value $u,v\in \mathbb{F}^n$0 (Lavi, 2018). In the nonnegative formulation, the $u,v\in \mathbb{F}^n$1 conjectured upper bound becomes

$u,v\in \mathbb{F}^n$2

where $u,v\in \mathbb{F}^n$3 and $u,v\in \mathbb{F}^n$4 (Lavi, 2018).

The paper also formulates conjectural singular-matrix bounds. For singular nonnegative $u,v\in \mathbb{F}^n$5,

$u,v\in \mathbb{F}^n$6

and for any diagonal product,

$u,v\in \mathbb{F}^n$7

When $u,v\in \mathbb{F}^n$8 has rank $u,v\in \mathbb{F}^n$9, these statements translate directly through $1$00 and $1$01, and are compatible with the exact rank-1 factorization (Lavi, 2018). The paper states that equality in the singular bounds is compatible with rank $1$02 only in low-dimensional or degenerate cases such as $1$03 or the presence of a zero row or column (Lavi, 2018).

4. Structured perturbations: $1$04, zeons, and Johnson-scheme submatrices

A different line of work studies permanents of matrices built from the identity and the all-ones matrix, notably

$1$05

Although $1$06 is generally not rank $1$07, its dependence on the rank-1 matrix $1$08 makes it a canonical rank-1 perturbation problem. The permanent admits the closed form

$1$09

and its exponential generating function is

$1$10

These formulas are identified with exponential moment polynomials in the zeon-algebra treatment (Feinsilver et al., 2017).

The same permanent also has a rencontres interpretation. For $1$11,

$1$12

where $1$13 and $1$14 denotes the number of derangements of $1$15 elements (Feinsilver et al., 2017). This connects the rank-1 perturbation $1$16 to classical permutation statistics through fixed points.

The zeon framework organizes these identities through zeon powers $1$17, whose entries are permanents of $1$18 submatrices. The trace identity

$1$19

is presented as a permanent analog of MacMahon’s Master Theorem (Feinsilver et al., 2017). For $1$20, every $1$21 submatrix is all ones, so all such submatrix permanents equal $1$22, and the closed form above follows immediately.

The high symmetry of $1$23 also produces a Johnson-scheme expansion for submatrix permanents of $1$24. If $1$25 are $1$26-subsets of $1$27, then

$1$28

so the submatrix permanent depends only on Johnson distance (Feinsilver et al., 2017). This uniformity is specific to the symmetric rank-1 matrix $1$29 and underlies the spectral decompositions in that paper.

The same framework yields combinatorial families called generalized derangements and generalized arrangements by specializing the parameters $1$30 and $1$31 (Feinsilver et al., 2017). A plausible implication is that rank-1 structure is not merely a degenerate algebraic case; it acts as the symmetric core from which richer permanent identities emerge when diagonal perturbations are introduced.

5. Approximation theory: Bethe and Sinkhorn permanents at rank 1

Low-rank permanent approximation provides another setting in which rank-1 matrices are analytically tractable. For an $1$32 nonnegative matrix $1$33, the Bethe and Sinkhorn permanents are defined through optimization over the Birkhoff polytope $1$34 of doubly stochastic matrices. With

$1$35

the definitions are

$1$36

$1$37

and

$1$38

They satisfy

$1$39

for all nonnegative $1$40 (Anari et al., 2020).

If $1$41 has non-negative rank $1$42, then the exact permanent is

$1$43

For the Sinkhorn objective,

$1$44

because $1$45 is doubly stochastic. Thus maximizing $1$46 reduces to maximizing entropy, whose maximizer is the uniform doubly stochastic matrix $1$47. Consequently,

$1$48

and

$1$49

Comparing with the exact permanent gives

$1$50

via Stirling’s approximation (Anari et al., 2020).

The same paper proves general approximation theorems for matrices of non-negative rank at most $1$51: $1$52 and similarly with $1$53 in place of the scaled Sinkhorn permanent (Anari et al., 2020). Specializing to $1$54 yields

$1$55

The paper also gives lower-bound constructions showing that for $1$56, including the all-ones matrix, polynomial-in-$1$57 gaps are unavoidable in worst case, so the $1$58 factor is tight up to constants (Anari et al., 2020).

Algorithmically, rank $1$59 is therefore an exceptional case. The exact permanent is immediate from the product formula, the scaled Sinkhorn permanent has a closed form, and the Bethe permanent is sandwiched between explicit quantities (Anari et al., 2020). This places rank-1 matrix permanents at the intersection of exact symbolic evaluation and convex-relaxation-based approximation theory.

6. Conceptual significance and limitations

Across these research directions, the principal invariant of a rank-1 matrix permanent is multiplicative rather than combinatorial: all permutation terms coincide. That collapse explains why the rank-1 case serves simultaneously as an extremal equality case, a normalization benchmark, and a solvable base case for broader low-rank theories.

In the $1$60 setting, rank $1$61 identifies the maximal permanent magnitude and the extremal equivalence class under standard transformations, with $1$62 as the canonical representative (Budrevich et al., 2018). In the nonnegative stochastic setting, rank $1$63 converts permanent maximization into a product maximization problem on the simplex, uniquely selecting the uniform matrix $1$64 (Lavi, 2018). In the structured family $1$65, the same rank-1 core yields explicit permanent formulas, Johnson-scheme expansions, and links to generalized derangements through the symmetry of $1$66 (Feinsilver et al., 2017). In approximation theory, rank $1$67 is the case where the exact permanent, Sinkhorn permanent, and asymptotic gap can all be written explicitly (Anari et al., 2020).

Several limitations are equally clear. The exact identity

$1$68

holds universally, but sharp global inequalities tied to rank can require stronger hypotheses, such as characteristic zero in the $1$69 extremal theory (Budrevich et al., 2018). Likewise, the clean stochastic maximizer $1$70 is specific to the $1$71 case in the bounded-rank conjectural framework; higher-rank conjectures involve block-diagonal composition matrices rather than a single outer product (Lavi, 2018). For general diagonal-plus-rank-1 perturbations $1$72, the zeon paper notes that closed forms comparable to the $1$73 case generally fail without additional symmetry (Feinsilver et al., 2017).

A plausible synthesis is that rank-1 matrix permanents are less a narrow special case than a structural boundary point of permanent theory. They mark the unique regime in which the permanent loses its usual dependence on permutation geometry and becomes completely determined by one multiplicative tensor factorization.