In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that every two different rows in a Hadamard matrix represent two perpendicular vector space|vectors, while in combinatorics|combinatorial terms, it means that every two different rows have matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The n-dimensional parallelepiped|parallelotope spanned by the rows of an n×n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so, is an extremal solution of Hadamard's maximal determinant problem.
Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller codes), and are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator.
In mathematics, an Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that every two different rows in a Hadamard matrix represent two perpendicular vector space|vectors, while in combinatorics|combinatorial terms, it means that every two different rows have matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The n-dimensional parallelepiped|parallelotope spanned by the rows of an n×n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so, is an extremal ...