Math Conjectures

Hadamard Matrix Conjecture

OPENLandmarkConjectureProposed 1893 · Full conjecture

Canonical statement

For every integer m≥1m\geq 1, there is a matrix H∈{−1,1}4m×4mH\in\{-1,1\}^{4m\times4m} such that
HHT=4mI4m. HH^{\mathsf T}=4mI_{4m}.
Here I4mI_{4m} is the 4m×4m4m\times4m identity matrix.
View source LaTeX
For every integer \(m\geq 1\), there is a matrix
\(H\in\{-1,1\}^{4m\times4m}\) such that
\[
  HH^{\mathsf T}=4mI_{4m}.
\]
Here \(I_{4m}\) is the \(4m\times4m\) identity matrix.
COMB-001:[Hadamard1893][HedayatSloaneStufken1999][KharaghaniTayfehRezaie2005]
CombinatoricsDiscrete geometry

Notes

The conjecture asserts that a Hadamard matrix — a square ±1\pm1 matrix HH with HHT=nInHH^{\mathsf T}=nI_n, so that its rows are pairwise orthogonal — exists for every order n=4mn=4m. The condition arose in Hadamard's 1893 work on the maximal determinant problem: among matrices with entries of modulus at most 11, Hadamard matrices attain the extremal determinant nn/2n^{n/2} [Hadamard1893].

An elementary argument shows that the order of a Hadamard matrix larger than 22 must be divisible by 44, so the conjecture claims this necessary condition is also sufficient. Many infinite families of constructions are known, from recursive doubling to number-theoretic and design-theoretic methods, and Hadamard matrices are closely tied to orthogonal arrays and experimental design [HedayatSloaneStufken1999]. These families cover a great many orders but leave gaps; a landmark sporadic construction was the order-428428 matrix of Kharaghani and Tayfeh-Rezaie [KharaghaniTayfehRezaie2005].

Orders divisible by 44 remain for which no Hadamard matrix is known, and no construction, or combination of constructions, is known to reach every multiple of 44. The conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.