Determinantal Hadamard Inequalities

• Determinantal Hadamard inequalities are bounds that relate a matrix’s determinant to functions of its diagonal entries or principal minors, achieving equality under specific structural conditions.
• They extend classical results like Hadamard and Fischer inequalities to broader matrix classes including k-positive, block-structured, and operator-valued matrices.
• Recent advances leverage hyperbolic polynomials and probabilistic methods to sharpen bounds, offering new insights for applications in combinatorics, random matrix theory, and PDE analysis.

A determinantal Hadamard inequality is, broadly, any inequality that bounds the determinant of a matrix—typically a positive semidefinite, Hermitian, or structured matrix—by the product, sum, or function of its diagonal entries, principal minors, or related quantities. Over more than a century, the classical inequality of Hadamard has been generalized in rich directions: for broader classes of matrices; for structured families of minors; for operator-algebraic and block-matrix frameworks; and in combinatorial and probabilistic settings. These inequalities form a foundational toolset in analysis, combinatorics, random matrix theory, operator algebras, and partial differential equations.

1. Foundational Inequalities: Classical Hadamard and its Extensions

The classical Hadamard determinant inequality asserts that for any Hermitian positive semidefinite $n \times n$ matrix $A = (a_{ij})$,

$|\det(A)| \leq \prod_{i=1}^n a_{ii},$

with equality if and only if $A$ is diagonal in some orthonormal basis. This inequality highlights the sub-multiplicative property of the determinant: the determinant cannot exceed the product of the norms of the row vectors, with the maximal value realized only in the case of mutual orthogonality.

Generalizing Hadamard's result, Fischer's inequality bounds the determinant of a principal submatrix by the product of determinants of block-diagonal components, and Koteljanskii's inequality synthesizes bounds involving overlapping principal minors. These inequalities have been unified and sharpened via sophisticated majorization techniques (Lin et al., 2020) and in the language of hyperbolic polynomials (Blekherman et al., 2021).

Block extensions such as Thompson's theorem assert that for a positive definite block matrix $A = [A_{ij}]$ (where each $A_{ij}$ itself is square),

$$\det A \leq \det\left([\det A_{ij}]_{i,j=1}^n \right)$$

with equality precisely when $A$ is block-diagonal (Lin et al., 2020).

2. Determinantal Hadamard Inequalities for General Classes of Matrices

Recent developments extend determinantal Hadamard inequalities to $k$-positive symmetric matrices, defined as matrices whose first $k$ elementary symmetric functions of eigenvalues are positive, a framework motivated by $k$-Hessian PDEs. For a $k$-positive $A\in M_n(\mathbb R)$,

$$S_k(A):=\sum_{|I|=k} \det(A_I)\leq S_k(\mathrm{diag}(A)) = s_k(a_{11},\dots,a_{nn})$$

where $A_I$ denotes the principal $k\times k$ submatrix and $s_k$ is the $k$th elementary symmetric function on the diagonal entries. Equality holds if and only if $A$ is diagonal. The classical case $k=n$ recovers the Hadamard inequality (Le, 2021).

In the context of structured arrays, the 2025 Teimoori–Khodakarami result for Pascal determinantal arrays $\mathrm{PD}_k(i,j)$ (the $k\times k$ minors of the Pascal matrix) establishes log-convexity inequalities: $$\mathrm{PD}_k(i,j)\mathrm{PD}_{k+2}(i,j) \geq \mathrm{PD}_{k+1}(i,j)^2$$ for all $k, i, j$, and for all $k\geq 0$, with equality only in trivial cases. This realizes a general determinantal Hadamard inequality at the level of discrete kernel minors, governed by the Dodgson condensation hierarchy and the log-concavity operator (Faal et al., 6 Dec 2025).

3. Hyperbolic and PSD-Stable Polynomials: Generalized Inequalities

The theory of hyperbolic polynomials and PSD-stable (“positive semidefinite-stable”) linear principal minor (lpm) polynomials provides a unifying language for determinantal inequalities. An lpm polynomial is a real linear combination of principal minors of a symmetric matrix: $P(X) = \sum_{I\subset [n]} a_I \det(X_I)$ such that $P$ is homogeneous and has no zeros on the Siegel upper half-space. For such a $P$, it is shown that for any orthogonal projection $\pi_\Pi(A)$ onto the block-diagonal part (partition $\Pi$), one has

$P(\pi_\Pi(A)) \geq P(A)$

for all $A$ in the hyperbolicity cone $H(P)$, which, in the case $P(X) = \det(X)$, recovers the cone of positive semidefinite matrices. In the “maximal reduction” to the diagonal, this yields

$P(\operatorname{diag}(A)) \geq P(A)$

for any PSD-stable lpm $P$ (Blekherman et al., 2021). This formalism generalizes Hadamard–Fischer inequalities to a wide class of hyperbolic operator settings.

4. Operator Algebraic and Block-Structured Generalizations

Finite von Neumann algebra methods and the Fuglede–Kadison determinant enable operator-valued versions of Hadamard and Fischer inequalities. Given a finite von Neumann algebra $\mathscr R$ with a faithful, normal tracial state $\tau$ and a $\tau$–preserving conditional expectation $\mathbb E$ onto a subalgebra $\mathscr S$,

$$\Delta\bigl(\mathbb E(A^{-1})^{-1}\bigr) \leq \Delta(A) \leq \Delta(\mathbb E(A))$$

for every positive invertible $A\in\mathscr R$, with equality iff $A$ lies in $\mathscr S$ (Nayak, 2017).

Block-matrix generalizations, including the Oppenheim and Oppenheim–Schur inequalities, establish lower bounds for determinants of the Hadamard (entrywise) and Khatri–Rao products of PSD block matrices. For positive definite $A^{(i)}\in M_n(M_{q_i}(\mathbb{C}))$,

$$\det\bigl(\ast_{i=1}^m A^{(i)}\bigr) \geq \prod_{i=1}^m \det A^{(i)} \prod_{\mu=2}^n \prod_{i=1}^m \det A^{(i)}_{\mu} - (m-1)$$

where $A^{(i)}_\mu$ is the $\mu\times \mu$ leading block principal minor (Li et al., 2020). Equality characterizations parallel the commutative (diagonal or block-diagonal) case.

5. Combinatorial and Probabilistic Determinantal Hadamard Bounds

Lower and upper bounds for determinants of $\{ \pm 1 \}$-matrices or $0$–$1$-matrices constitute a substantial branch of the field. For the maximal determinant $D(n)$ of $n\times n$ $\{\pm1\}$-matrices, Hadamard’s upper bound is $D(n)\leq n^{n/2}$, realized precisely (conjecturally) by Hadamard matrices of order $n \equiv 0\pmod4$. Recent probabilistic constructions yield uniform lower bounds: for $n = h + d$, where $h$ is the largest Hadamard order $\le n$,

$R(n) = \frac{D(n)}{n^{n/2}} > 0.07 \cdot (0.352)^d$

and, for fixed $d$ and large $n$,

$R(n) \geq \left( \frac{2}{\pi e} \right)^{d/2}$

with the latter conjectured to hold for all $n$ (Brent et al., 2012). The techniques crucially utilize Schur complements, moment estimates, and combinatorial identities such as binomial sum evaluations (Brent et al., 2013).

Improved upper bounds for sparse $0$–$1$ matrices with row sums $k$ show that for any fixed $k$,

$$\max_{A\in R(n,k)} |\det(A)| \leq c_k^n,\quad c_k<\sqrt{k}$$

thus demonstrating exponential improvements over the classical Hadamard bound for sparse regimes (Scheinerman, 2019).

Parallel lower bounds, both conditional and unconditional on the Hadamard conjecture, derive from bordering and minor techniques, as well as Jacobi's determinant identity, yielding almost-optimal rates for $R(n)$ and clarifying the gap between upper and lower bounds (Brent et al., 2012).

6. Majorization, Sharpened Inequalities, and Equality Cases

Contemporary approaches leverage majorization theory to yield sharpened forms of determinantal Hadamard inequalities. For Hermitian PSD matrices, Zhang and Yang (1997) and subsequent work demonstrated that

$$\det A + \sum_{i=1}^n |a_{i, \omega(i)}| \leq \prod_{i=1}^n a_{ii}$$

for all nontrivial permutations $\omega$, with equality if and only if every $2$-cycle in $\omega$ corresponds to a pair of collinear columns in $A$ (Lin et al., 2020). These refinements clarify the structure of extremal matrices for which Hadamard-type inequalities are tight, complementing characterizations via block-structure and permutation invariance.

In operator algebraic extensions, equality occurs precisely when all block or entrywise off-diagonal elements vanish with respect to the chosen partition or expectation.

7. Open Directions and Contextual Significance

Determinantal Hadamard inequalities interface with total positivity, log-concavity hierarchies, and the theory of hyperbolic polynomials, admitting combinatorial, analytic, algebraic, and operator-theoretic interpretations. Active directions include operator-valued determinants, inequalities for broader classes of homogeneous symmetric hyperbolic polynomials, matrix function inequalities (e.g. for Schur, Oppenheim, or Alexandrov-Fenchel-type frameworks), extensions to PDEs via $k$-Hessian operators, and optimality/tightness results in the sparse or block-structured matrix settings (Blekherman et al., 2021, Le, 2021, Faal et al., 6 Dec 2025).

Integration of probabilistic techniques with analytic determinant bounds continues to refine our understanding of extremal structures in combinatorial matrix theory. Recent work highlights the effectiveness of techniques such as Schur complements, hyperbolic programming, and randomization in yielding both upper and lower bounds near fundamental theoretical limits.

Determinantal Hadamard inequalities thus provide a unifying backbone for diverse applications, from algebraic combinatorics to functional analysis and discrete optimization, with ongoing generalizations driven by operator algebra, matrix analysis, and convex geometry.