Permanent of PSD Matrices

• Permanent of PSD matrices is defined as the sum over all permutations of matrix entries, capturing key combinatorial and algebraic invariants.
• The permanental rank-nullity theorem uniquely relates the size of the largest nonzero permanent submatrix to the multiplicity of zero in the permanental polynomial, paralleling the classical rank-nullity theorem.
• Computational approaches leverage approximation algorithms and convex relaxations to tackle the #P-complete challenge of computing permanents in positive semidefinite matrices.

A positive semidefinite (PSD) matrix is a symmetric or Hermitian matrix with all eigenvalues nonnegative, a property central to mathematical physics, optimization, quantum information theory, and combinatorics. The permanent of a matrix, while structurally similar to the determinant, lacks sign alternations and is computationally #P-hard to evaluate. The paper of the permanent of positive semidefinite matrices spans topics from structural characterization and algebraic invariants to computational approaches and approximation algorithms, with connections to deep combinatorial and algebraic principles.

1. Definition and Basic Properties

Given $A \in \mathbb{C}^{n \times n}$ (or $\mathbb{R}^{n \times n}$), the permanent is defined as

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

where $S_n$ is the symmetric group on $n$ elements. If $A$ is positive semidefinite (denoted $A \succeq 0$), all principal minors are nonnegative and all eigenvalues are real and nonnegative.

For PSD matrices, several important properties distinguish the permanent from the determinant:

• Monotonicity: If $0 \preceq A \preceq B$ then $\operatorname{per}(A) \leq \operatorname{per}(B)$.
• Lower Bounds: For any PSD matrix $A$,

$\operatorname{per}(A) \geq \prod_{i=1}^n a_{ii}$

due to Schur's theorem for nonnegative definite matrices.

• Factorization Structure: Every principal submatrix of a PSD matrix is also PSD, which constrains the support structure for nonvanishing permanents (Pant et al., 2 Jul 2025).

2. Permanental Rank, Nullity, and Analog of Rank-Nullity Theorem

For a matrix $A$, the permanental rank $\rho_{\mathrm{per}}(A)$ is the largest $k$ such that $A$ contains a $k \times k$ (principal) submatrix with nonzero permanent. The permanental nullity $\eta_{\mathrm{per}}(A)$ is defined as the multiplicity of $x = 0$ as a root of the permanental polynomial $\pi(A, x) = \operatorname{per}(x I - A)$. For PSD matrices, the following identity holds (Pant et al., 2 Jul 2025): $$\rho_{\mathrm{per}}(A) + \eta_{\mathrm{per}}(A) = n.$$ This result is formally analogous to the classical rank-nullity theorem for determinantal rank:

• The permanental rank is characterized by the existence of large principal submatrices with nonzero permanent.
• The nullity counts the degree of vanishing at $x = 0$ in $\pi(A, x)$ and, for PSD matrices, is precisely the complement of the largest nonvanishing-permanent submatrix size.

This equality, which also holds for nonnegative symmetric and $(-1,0,1)$-balanced symmetric matrices, ties combinatorial structures (cycle covers in graphs) to algebraic invariants.

3. Structural, Spectral, and Graph-Theoretic Interpretations

In the case of PSD matrices, structural insights align with graph-theoretic representations:

• Nonzero permanents correspond to cycle covers in the associated (support) graph.
• The permanental rank counts the maximum number of vertices that can be "covered" by vertex-disjoint cycles.
• The remaining vertices correspond to the permanental nullity.

For a PSD matrix, since all principal submatrices are also PSD, submatrix permanents are intrinsically linked to the structure of the whole matrix, leading to strict relationships between algebraic, spectral, and combinatorial characteristics.

4. Computational Complexity and Algorithmic Considerations

While classical rank computations (via determinants) can be performed in polynomial time, the permanent is known to be #P-complete to compute, even for nonnegative matrices. The permanental rank and permanental nullity, while theoretically well-defined, are computationally intractable in general:

• Evaluating $\operatorname{per}(A)$ for general or even PSD matrices is #P-hard.
• Computing $\rho_{\mathrm{per}}(A)$ or $\eta_{\mathrm{per}}(A)$ is at least as hard as determining whether a principal submatrix has nonzero permanent.
• For special classes (e.g., block-diagonal, highly structured, or small matrices), exact computation or sharp approximation is possible (Anari et al., 2017, Yuan et al., 2020).

Various approximation algorithms and relaxations have been developed to estimate the permanent of PSD matrices within exponential or subexponential multiplicative factors, employing convex optimization, randomized rounding, and quantum-inspired methods (Anari et al., 2017, Yuan et al., 2020, Chakhmakhchyan et al., 2016). However, no general polynomial-time algorithm exists for exact computation.

5. Comparison with Determinantal Rank-Nullity and Invariants

The classical rank-nullity theorem states, for a symmetric $n \times n$ matrix $A$, that the (determinantal) rank plus the nullity (geometric multiplicity of zero eigenvalue) equals $n$. In the case of the permanent, the analogous statement

$$\rho_{\mathrm{per}}(A) + \eta_{\mathrm{per}}(A) = n$$

requires the PSD condition (or, more generally, nonnegative symmetric or $(-1,0,1)$-balanced symmetric matrices). The determinant is multiplicative over block-diagonal decompositions and products, whereas the permanent lacks full multiplicativity; the structure imposed by PSD, however, ensures strict bounds and equalities not generally available for arbitrary matrices.

Table: Comparison of Determinantal and Permanental Analogues for PSD Matrices

Invariant	Determinant	Permanent
Rank	Size of largest nonsingular submatrix	Size of largest submatrix with nonzero permanent
Nullity	Zero eigenvalue multiplicity	Multiplicity of zero root in permanental poly.
Rank + Nullity	$n$	$n$ (for PSD matrices)
Computability	Polynomial	#P-complete

6. Implications and Applications

The permanental rank-nullity theorem has several far-reaching consequences:

• Structural Analysis: Confirms perfect complementarity between the largest submatrix with nonzero permanent and the degree of vanishing in the permanental polynomial, leading to insight into the combinatorial complexity and "cycle structure" of PSD matrices.
• Spectral Theory: Offers an algebraic invariant that parallels the spectrum, but is sensitive to combinatorics of cycle covers rather than to sign structure or orthogonality.
• Complexity Theory: Highlights the contrast between algebraic and combinatorial invariants: while both satisfy elegant equalities, permanental quantities remain intractable in general, suggesting a "hidden complexity" in PSD-matrix structure.
• Broader Classes: The result extends to nonnegative symmetric matrices and $(-1,0,1)$-balanced symmetric matrices, reflecting a deep algebraic-combinatorial connection.
• Graph Theory: The result links cycle covers in graphs (which correspond to nonvanishing submatrix permanents) with algebraic invariants and is relevant for the analysis of matching polynomials and related objects.

7. Open Problems and Research Directions

The permanental rank-nullity theorem stimulates further inquiry in several directions:

• Efficient Detection: For which subclasses of PSD matrices can the permanental rank or nullity be computed efficiently? Are there sharp characterizations that leverage particular block structures, sparsity, or symmetries?
• Relaxations and Approximations: Can approximation algorithms (deterministic or randomized) for the permanent be effectively harnessed to estimate permanental rank and nullity on large-scale instances?
• Applications in Quantum Information: Since PSD matrices encode quantum states and correlations, understanding permanental invariants may yield new insights into quantum complexity, entropy, and correlation structure.
• Extensions to Other Matrix Classes: The result's extension to $(-1,0,1)$-balanced symmetric matrices hints at its applicability to signed graphs and broader combinatorial classes.

The exact connection between permanental rank, combinatorial graph covers, and algebraic spectral invariants underscores the intersection of combinatorics, matrix analysis, and computational complexity in the paper of PSD matrices.