Least change in the Determinant or Permanent of a matrix under perturbation of a single element: continuous and discrete cases

We formulate the problem of finding the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that the randomly chosen elements have a fixed probability of being non-zero. Also, we show that the procedure for finding the probability that the determinant undergoes the least change depends on whether the random variables for the matrix elements are continuous or discrete.

💡 Research Summary

The paper investigates the probability that perturbing a single entry of a random matrix causes the smallest possible change in the matrix’s determinant (or permanent). The authors consider (n + 1) × (n + 1) matrices Mₙ₊₁ and focus on the cofactor Mᵢⱼ associated with the entry mᵢⱼ that is to be perturbed. Since the determinant expansion along row i contains the term mᵢⱼ Mᵢⱼ as the only occurrence of mᵢⱼ, the magnitude of the change caused by perturbing mᵢⱼ is proportional to |Mᵢⱼ| = |det Sₙ|, where Sₙ is the n × n sub‑matrix obtained by deleting row i and column j. Consequently, the problem reduces to finding the probability that |det Sₙ| is as small as possible.

Three families of random matrices are defined, each characterized by a fixed probability r (0 < r < 1) that a variable entry is non‑zero:

1. Type Aₙ – every entry is a variable; each entry equals zero with probability 1 − r and a non‑zero value with probability r.
2. Type Bₙ – all diagonal entries are 1 except possibly the (1,1) entry, which is variable; all off‑diagonal entries are of type Aₙ.
3. Type Cₙ – all diagonal entries are fixed at 1; all off‑diagonal entries are of type Aₙ.

The analysis distinguishes between continuous and discrete distributions for the variable entries. In the continuous case, the probability of any specific non‑zero value is zero, so the only way to achieve the minimal change is for det Sₙ to be exactly zero (for types Aₙ and Bₙ) or exactly one (for type Cₙ).

To handle the combinatorial aspect, each random matrix Sₙ is mapped to a binary matrix ˜Sₙ by setting ˜sᵢⱼ = 1 if the corresponding variable entry is non‑zero and ˜sᵢⱼ = 0 otherwise. The permanent of ˜Sₙ, per ˜Sₙ, is then examined. The authors prove the following equivalences:

• For types Aₙ and Bₙ, det Sₙ = 0 ⇔ per ˜Sₙ = 0.
• For type Cₙ, det Sₙ = 1 ⇔ per ˜Sₙ = 1.

Thus the original probability problem becomes a counting problem for binary matrices with a prescribed permanent value.

Let m denote the total number of variable entries in ˜Sₙ and let i be the number of 1’s among them. For a given i, denote by Eₙ(i) the number of binary matrices of the appropriate type that have per ˜Sₙ = u (u = 0 for Aₙ and Bₙ, u = 1 for Cₙ) and exactly i ones. Because each variable entry independently equals 1 with probability r and 0 with probability 1 − r, the probability that a particular matrix has exactly i ones is rⁱ(1 − r)^{m‑i}. Summing over all admissible i yields

P(per ˜Sₙ = u) = ∑_{i=0}^{i_max} Eₙ(i) rⁱ (1 − r)^{m‑i}.

The paper derives explicit expressions for m, i_max, and the minimal number j_min of all‑zero rows (or columns) that force the permanent to be zero. For type Aₙ, j_min = n (any full row or column of zeros), giving i_max = n² − n; for type Bₙ, j_min = n as well, giving i_max = (n − 1)²; for type Cₙ, the condition per ˜Sₙ = 1 corresponds to the binary matrix representing an acyclic directed graph (digraph) on n vertices, so i_max = (m − j_min) = (n² − n)/2.

The authors introduce three integer sequences to enumerate the relevant matrices:

• Fₙ(i) for type Aₙ (per ˜Aₙ = 0).
• Gₙ(i) for type Bₙ (per ˜Bₙ = 0).
• Hₙ(i) for type Cₙ (per ˜Cₙ = 1).

For small n (1 ≤ n ≤ 5) the paper lists the values of these sequences, showing that Fₙ(i) coincides with OEIS A08867, while Gₙ(i) appears to be a new sequence. The sequence Hₙ(i) is shown to be equal to the number of acyclic digraphs with i edges on n labeled vertices, a well‑studied combinatorial object (related to OEIS A003024).

Consequently, the probability that perturbing a single entry yields the least possible change in the determinant is expressed as a polynomial in r whose coefficients are given by the above sequences. For example, for type Aₙ the probability is

P_Aₙ(r) = ∑_{i=0}^{n²‑n} Fₙ(i) rⁱ (1 − r)^{n²‑i}.

Analogous formulas hold for types Bₙ and Cₙ with Gₙ(i) and Hₙ(i).

The paper also discusses the discrete case, where the random variable can take a finite set of non‑zero values with positive probability. In that scenario, the event det Sₙ = 0 is no longer the only way to achieve minimal change; other non‑zero determinants may also have non‑zero probability. The authors outline how the same combinatorial framework can be adapted, but the detailed enumeration is left for future work.

In summary, the work provides a clear reduction of a matrix‑sensitivity problem to counting binary matrices with prescribed permanents, connects the counting to well‑known combinatorial structures (full zero rows/columns and acyclic digraphs), and supplies explicit probability formulas for three natural families of random matrices. The results bridge linear algebra, probability theory, and enumerative combinatorics, and they open avenues for further investigation of larger matrices, asymptotic behavior, and extensions to other perturbation models.