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[parent] proof that Hadamard matrix has order 1 or 2 or 4n (Proof)

Let $m$ be the order of a Hadamard matrix. The matrix $[1]$ shows that order 1 is possible, and the parent entry has a $2 \times 2$ 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 submatrix of one row and all 1's and $N$ denotes a submatrix of one row and all $-1$ s.

$$\begin{matrix} \begin{matrix} x \quad &\quad y & \quad z & \quad w \end{matrix} & \begin{matrix} \quad \end{matrix} \\ \left[ \begin{matrix} \overbrace{P} & \overbrace{P} & \overbrace{P} & \overbrace{P} \\ P & P & N & N \\ P & N & P & N \\ \end{matrix} \right] \end{matrix} $$

Since the rows are orthogonal and there are $m$ columns we have

$\begin{cases} x + y + z +w &= m \\ x + y - z - w &= 0 \\ x - y + z -w &= 0 \\ x - y - z + w &= 0. \end{cases}$
Adding the 4 equations together we get $$ 4x = m. $$ so that $m$ must be divisible by 4.




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Cross-references: divisible, equations, orthogonal, submatrix, columns, row, matrix, Hadamard matrix, order

This is version 7 of proof that Hadamard matrix has order 1 or 2 or 4n, born on 2007-03-18, modified 2007-03-20.
Object id is 9095, canonical name is ProofThatHadamardMatrixHasOrder12Or4n.
Accessed 1119 times total.

Classification:
AMS MSC15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 05B20 (Combinatorics :: Designs and configurations :: Matrices )

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