KnowraHadamard conjectureHadamard conjectureThe Hadamard conjecture states that a Hadamard matrix exists for every positive order divisible by four.BriefConnectHadamard matrix: A square matrix with entries ±1 whose distinct rows are mutually orthogonal. These matrices are the objects whose existence the conjecture predicts at every order divisible by four.Sylvester Hadamard matrix: A recursively defined Hadamard matrix of order a power of two, beginning with the order-two matrix. Its recursion demonstrates a simple mechanism behind the existence of matrices at infinitely many orders.Hadamard matrix of order 12: A Hadamard matrix with twelve rows and twelve columns. This small non-power-of-two example illustrates existence beyond Sylvester's basic family.Hadamard matrix existence problem: The problem of determining exactly which positive integers are orders of Hadamard matrices. The conjecture is the strongest simple proposed answer to this broader classification problem.Orthogonal matrix: A real square matrix whose transpose is its inverse, so its columns form an orthonormal basis. Scaling a Hadamard matrix by the reciprocal square root of its order produces an orthogonal matrix.Kronecker product: A matrix operation that replaces each entry of one matrix with a scaled copy of another matrix. The Kronecker product of Hadamard matrices yields another Hadamard matrix, multiplying their orders.Hadamard matrix of order 20: A Hadamard matrix with twenty rows and twenty columns. Its existence is a concrete case not supplied by the powers-of-two construction.Hadamard matrix enumeration: The study of counting Hadamard matrices up to specified equivalence relations. Enumeration can verify and classify examples, but cannot by itself prove existence at every order.Orthogonality: The condition that vectors have zero inner product, making them perpendicular in an inner-product space. Mutually orthogonal rows are the defining structural condition in a Hadamard matrix.Williamson construction: A method for constructing Hadamard matrices from four commuting symmetric circulant matrices. It is a structured construction that settles existence for some orders beyond Sylvester's family.Show all 20Linked from 1 pageJacques HadamardRelated: The conjecture grew from the matrices associated with Hadamard’s determinant bound and remains unresolved.