Editing Hadamard matrix (section)

{{Short description|Mathematics concept}}
{{Use dmy dates|date=February 2023}}
[[File:HadamardConjectureMIT.png|thumb|[[Gilbert Strang]] explains the Hadamard conjecture at [[MIT]] in 2005, using Sylvester's construction.]]
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 [[geometry|geometric]] terms, this means that each pair of rows in a Hadamard matrix represents two [[perpendicular]] [[vector (mathematics and physics)|vector]]s, while in [[combinatorics|combinatorial]] terms, it means that each pair of rows has 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|parallelotope]] spanned by the rows of an ''n''&thinsp;×&thinsp;''n'' Hadamard matrix has the maximum possible {{nowrap|''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 [[matrix (mathematics)|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 code]]s), and are also used in [[balanced repeated replication]] (BRR), used by [[statistician]]s to estimate the [[variance]] of a [[parameter]] [[estimator]].