====== Hadamard conjecture ======

>A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with {{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline1.svg?21x21|-1}}s are colored white. Therefore, the {{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline2.svg?35x21|n×n}} Hadamard matrix {{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline3.svg?20x21|H_n}} must have {{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline4.svg?80x21|n(n-1)/2}} white squares ({{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline5.svg?21x21|-1}}s) and {{https://mathworld.wolfram.com/images/equations/HadamardMatrix/Inline6.svg?80x21|n(n+1)/2}} black squares (1s)."\\ \\ Source : [[https://mathworld.wolfram.com/HadamardMatrix.html|Wolfram Mathworld]]

An 'open question' remains regarding the matrices : the //Hadamard conjecture// proposes that a Hadamard matrix of order 4//k// exists for every positive integer //k. //

Put another way,

>An n × n matrix// H //with all its entries +1 and −1 is Hadamard //if HH′ = nI.// It is well known that //n// must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every// n// which is a multiple of 4."\\ \\ Source : [[https://projecteuclid.org/journals/annals-of-statistics/volume-6/issue-6/Hadamard-Matrices-and-Their-Applications/10.1214/aos/1176344370.full|Annals of Statististics 6(6): 1184-1238]]{{:oa_padlock_grn.png?16}}

This conjecture has neither been proved or disproved.