proof that Hadamard matrix has order 1 or 2 or 4n

Let m be the order of a Hadamard matrixMathworldPlanetmath. The matrix [1] shows that order 1 is possible, and the entry has a 2×2 Hadamard matrix , so assume m>2.

We can assume that the first row of the matrix is all 1’s by multiplying selected columns by -1. Then permute columns as needed to arrive at a matrix whose first three rows have the following form, where P denotes a submatrixMathworldPlanetmath of one row and all 1’s and N denotes a submatrix of one row and all -1’s.


Since the rows are orthogonalMathworldPlanetmath and there are m columns we have


Adding the 4 equations together we get


so that m must be divisible by 4.

Title proof that Hadamard matrix has order 1 or 2 or 4n
Canonical name ProofThatHadamardMatrixHasOrder1Or2Or4n
Date of creation 2013-03-22 16:50:56
Last modified on 2013-03-22 16:50:56
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 10
Author Mathprof (13753)
Entry type Proof
Classification msc 05B20
Classification msc 15-00