Approximate Hadamard Matrices Overview

• Approximate Hadamard matrices are n×n ±1 matrices whose singular values lie within [c√n, C√n], generalizing classical Hadamard matrices.
• They are constructed via prime-size block methods, arithmetic decompositions, and randomized perturbations to ensure well-conditioned spectral properties.
• These matrices have practical uses in numerical linear algebra, signal processing, and coding, providing robust preconditioning and near-orthogonal designs.

An approximate Hadamard matrix is an $n \times n$ matrix with entries in $\{\pm1\}$ whose singular values all lie in a window $[c\sqrt{n}, C\sqrt{n}]$ for some absolute constants $0 < c < C < \infty$ independent of $n$. These matrices substantially generalize classical Hadamard matrices, which correspond to the exact case $c = C = 1$. The concept arises naturally both from longstanding questions in combinatorial design theory and from practical requirements in signal processing and random matrix theory when exact orthogonality is unattainable.

1. Formal Definitions and the Hadamard Problem

Let $A \in \{\pm1\}^{n \times n}$ and denote its singular values by $\sigma_1(A) \geq \ldots \geq \sigma_n(A) \geq 0$. The condition number is $\kappa(A) = \sigma_1(A)/\sigma_n(A)$. A classical (exact) Hadamard matrix satisfies $A A^T = n I_n$, so all singular values are $\sqrt{n}$ and $\kappa(A) = 1$. The Hadamard conjecture posits the existence of such matrices for all $n$ divisible by 4.

In dimensions where exact Hadamard matrices provably do not exist or their existence is open, the approximate Hadamard matrix relaxes the requirement, asking that

$$c\sqrt{n} \leq \sigma_n(A) \leq \sigma_1(A) \leq C\sqrt{n}$$

for constants independent of $n$. This is equivalent to the operator norm inequality

$$c\sqrt{n} \|x\|_2 \leq \|Ax\|_2 \leq C\sqrt{n}\|x\|_2 \qquad \forall x \in \mathbb{R}^n$$

and to a uniformly bounded condition number $\kappa(A) \leq C/c$ across all $n$ (Steinerberger, 20 Feb 2024, Dong et al., 2022, Alexeev et al., 18 Nov 2025).

2. Existence Theorems and Construction Strategies

The main existence theorem, due to Dong and Rudelson, asserts that for every $n \geq 1$, there exists a matrix $A \in \{\pm1\}^{n \times n}$ with all singular values in a fixed $[c\sqrt{n}, C\sqrt{n}]$ window, for absolute $0 < c < C < \infty$ (Steinerberger, 20 Feb 2024). The construction can be summarized as follows:

• Prime-size blocks: For each odd prime $q$, a $q \times q$ matrix whose entries are determined by the Legendre symbol $\left(\frac{i-j}{q}\right)$, exploits classical Gauss sum bounds to guarantee flatness of the Fourier spectrum and singular values in $[c\sqrt{q}, C\sqrt{q}]$.
• Gluing via arithmetic decompositions: By Vinogradov's three-primes theorem, any large $n$ can be written as a sum of a bounded number of primes, allowing assembly of blocks while controlling spectral deviations.
• Approximate Hadamard blocks: A "gluing" procedure pads and signs the blocks to form an $n\times n$ matrix, preserving the spectral window (Steinerberger, 20 Feb 2024, Dong et al., 2022).
• Random noise augmentation: In (Dong et al., 2022), a small random Rademacher perturbation can be incorporated to fill out the full dimension and further concentrate the spectrum.

An alternative, nonconstructive proof uses "flat" orthogonal matrices—orthogonals with entries as close as possible to $1/\sqrt{n}$—and a subsequent randomized rounding to $\pm1$, with matrix Bernstein concentration controlling the eigenvalue spread. For all large $n$, this yields

$\kappa(n) \leq 1 + n^{-\alpha}$

for some $\alpha > 0$, implying the minimal achievable condition number among all $\{\pm1\}$-matrices converges to $1$ as $n \to \infty$ (Alexeev et al., 18 Nov 2025).

3. Explicit Infinite Families of Approximations

Several explicit families achieve $\kappa(A) = 1 + O(n^{-1/2})$ or $\sqrt{2} + o(1)$:

Construction	Parameter Regime	Condition Number
Symmetric conference	$n \equiv 2 \pmod 4$	$1 + O(n^{-1/2})$
Barba matrices	$n \equiv 1 \pmod 4,\ n = 2(q^2 + q) + 1$	$\sqrt{2} + o(1)$
SDS block construction	$n \equiv 2 \pmod 4,\ \frac{n}{2}=q^2 + q + 1$	$\sqrt{2} + o(1)$
Flat Littlewood Circulants	All $n$	$O(1)$, non-optimal

For each construction, the design ensures that $A^T A$ is close to $n I$ in the spectral sense, but not exactly, so exact orthogonality is replaced by approximate spectral flatness:

• Symmetric conference matrices $C$ with $C_{ii}=0$, $C_{ij}=\pm1$, $C^T C = (n-1)I$ yield $A = C+I_n \in \{\pm1\}^{n\times n}$ with $\kappa(A) = \frac{\sqrt{n-1}+1}{\sqrt{n-1}-1}$ (Alexeev et al., 18 Nov 2025).
• Barba matrices satisfy $A^T A = (n-1)I_n + J_n$, producing $\kappa(A) = \sqrt{\frac{2n-1}{n-1}}$ (Alexeev et al., 18 Nov 2025).
• Supplementary-difference-set (SDS) constructions combine two circulant blocks $R,S$ obeying $R^T R + S^T S = (n-2)I + 2J$ to build block matrices $A$ with controlled singular spectrum (Alexeev et al., 18 Nov 2025).
• Circulant flat matrices: Via existence of flat Littlewood polynomials (i.e., $\pm1$-polynomials $p(z)$ such that $|p(e^{it})| \in [c\sqrt{n}, C\sqrt{n}]$ for all $t$), one constructs circulant $A = \mathrm{circ}(a_0,\ldots,a_{n-1})$ with singular values bounded in $[c\sqrt{n}, C\sqrt{n}]$ (Steinerberger, 20 Feb 2024).

4. Circulant Approximate Hadamard Matrices and Flat Littlewood Polynomials

A circulant matrix $A = \mathrm{circ}(a_0, ..., a_{n-1})$ has eigenvalues given by evaluation of the generating polynomial $p(z) = a_0 + a_1 z + \ldots + a_{n-1} z^{n-1}$ at $n$th roots of unity. If $p$ is "flat" on the unit circle, meaning $|p(e^{it})| \in [c \sqrt{n}, C\sqrt{n}]$ for all $t \in [0,2\pi]$, then $A$ is a circulant approximate Hadamard matrix with the desired spectral window.

Balister, Bollobás, Morris, Sahasrabudhe, and Tiba have established the unconditional existence of such flat Littlewood polynomials for every $n$ (Steinerberger, 20 Feb 2024). Thus, circulant $\{\pm1\}$-matrices matching the Dong–Rudelson criteria exist in all dimensions. This provides the first universal construction of well-conditioned circulant $\{\pm1\}$-matrices with operator norm and minimal singular value both proportional to $\sqrt{n}$.

This result does not contradict Ryser's conjecture, which asserts the non-existence of exact circulant Hadamard matrices for $n>4$, because these approximate versions only require the singular values to be well-separated but not equal (Steinerberger, 20 Feb 2024). Moreover, quantitative conjectures predict a polynomial spectral gap away from the optimal value in the non-Hadamard orders.

5. Relationship to Schur Norms, Almost Hadamard Matrices, and Design Theory

The theory of approximate Hadamard matrices intersects with other relaxations in the literature:

• Schur norm maximization: The maximal Schur (Hadamard) norm of an $n\times n$ $\{\pm1\}$-matrix is $\sqrt{n}$ if and only if it is a true Hadamard. For non-Hadamard orders, maximizing the Schur norm yields "almost Hadamard matrices," which closely approximate the ideal but always fall short (Holbrook et al., 2022). The best-known examples (e.g., for $n=5,6,7$) achieve $\|M\|_S \approx 0.97-0.98 \times \sqrt{n}$.
• Almost Hadamard matrices (AHM): Defined as orthogonal matrices whose entries are as close to $\pm1$ as possible, and which are local maxima of the $1$-norm on $O(N)$ (Banica et al., 2012, Banica et al., 2012, Banica et al., 2017). These are highly structured, admit explicit classification results for certain patterns and circulant cases, but generally do not achieve the spectral flatness required of approximate Hadamard matrices in the Dong–Rudelson or Alexeev–Jasper–Mixon sense.
• Pseudo-Hadamard and block-design constructions: Families derived from removing rows/columns or "thinning" classical Hadamard matrices yield matrices with almost-orthogonal rows, relevant for systematic construction in missing dimensions (Sharipov, 2021).

The design perspective (incidence matrices from block designs, conference matrices, Barba matrices) provides infinite families of explicit near-optimal examples, which are especially valuable in cases where general probabilistic or number-theoretic constructions are available only nonconstructively (Alexeev et al., 18 Nov 2025).

6. Applications and Open Questions

Approximate Hadamard matrices enable robust and nearly orthogonal sign-preconditioned matrix ensembles for applications where only $\{\pm1\}$ entries are permissible:

• Numerical linear algebra: They can be deployed as well-conditioned preconditioners, random projections, and for sketching algorithms needing bounded isometry constants (Alexeev et al., 18 Nov 2025).
• Signal processing and coding: Near-orthogonal $\{\pm1\}$ matrices preserve signal energy and minimize cross-correlation in communication protocols, generalizing the Hadamard–Walsh approaches.
• Frame theory and combinatorics: They contribute to the design of new frame families and measuring ensembles where exact orthogonality is too restrictive (Dong et al., 2022).

Open questions include optimizing the exponent $\alpha$ in condition number convergence to its tightest possible value, extending explicit families to new congruence classes, finding odd-order constructions with condition number below $\sqrt{2}$, and establishing more refined quantitative versions of the Ryser conjecture in the approximate regime (Steinerberger, 20 Feb 2024, Alexeev et al., 18 Nov 2025).

7. Comparison with Exact Hadamard and Limiting Behavior

Approximate Hadamard matrices do not close the Hadamard conjecture but fill an important gap by providing well-conditioned $\{\pm1\}$ matrices in all dimensions, with condition numbers arbitrarily close to $1$ for large $n$—matching exact Hadamard matrices asymptotically (Alexeev et al., 18 Nov 2025). This universality is achieved at the cost of perfect orthogonality, which remains exclusive to exact Hadamard matrices; the quantitative gap is well-characterized and diminishing as matrix size grows. The structure theory unifies probabilistic, algebraic, and analytic tools, promising further progress in both construction and characterization.