Hadamardsches

Hadamardsches refers to concepts and objects named after the French mathematician Jacques Hadamard. In mathematics, the Hadamard matrix is a specific type of square matrix. A Hadamard matrix of order n is an n x n matrix H with entries +1 or -1 such that HH^T = nI_n, where H^T is the transpose of H and I_n is the n x n identity matrix. Equivalently, the rows of a Hadamard matrix are mutually orthogonal, as are the columns. The order n of a Hadamard matrix must be 1, 2, or a multiple of 4. It is an open problem whether Hadamard matrices exist for all orders that are multiples of 4.

The Hadamard product, also known as the element-wise product or Schur product, is another operation named after Hadamard. For two matrices A and B of the same dimensions, their Hadamard product, denoted A ∘ B, is a matrix C where each element c_{ij} is the product of the corresponding elements of A and B, i.e., c_{ij} = a_{ij}b_{ij}. This operation differs from the standard matrix multiplication.

The Hadamard inequality, also known as Hadamard's maximal determinant problem, provides an upper bound for the determinant of a real matrix. For a real n x n matrix A with entries |a_{ij}|, the absolute value of the determinant is bounded by the product of the Euclidean norms of its column vectors: |det(A)| <= \prod_{j=1}^n ||a_j||_2. Equality holds if and only if the columns of A are mutually orthogonal.

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