proof that Hadamard matrix has order 1 or 2 or 4n
Let be the order of a Hadamard matrix. The matrix shows that order 1 is possible, and the entry has a Hadamard matrix , so assume .
We can assume that the first row of the matrix is all 1’s by multiplying selected columns by . Then permute columns as needed to arrive at a matrix whose first three rows have the following form, where denotes a submatrix of one row and all 1’s and denotes a submatrix of one row and all ’s.
Since the rows are orthogonal and there are columns we have
Adding the 4 equations together we get
so that must be divisible by 4.
|Title||proof that Hadamard matrix has order 1 or 2 or 4n|
|Date of creation||2013-03-22 16:50:56|
|Last modified on||2013-03-22 16:50:56|
|Last modified by||Mathprof (13753)|