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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.

$\displaystyle \begin{matrix} \begin{matrix} x \quad &\quad y & \quad z & \quad ... ...ce{P} \ P & P & N & N \ P & N & P & N \ \end{matrix} \right] \end{matrix}$

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

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



"proof that Hadamard matrix has order 1 or 2 or 4n" is owned by Mathprof.
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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 586 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 05B20 (Combinatorics :: Designs and configurations :: Matrices )
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