ProofThatHadamardMatrixHasOrder12Or4n 2007-03-18 17:18:32 2007-03-20 09:42:13 Proof Hadamard matrix 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $m$ be the order of a Hadamard matrix. The matrix $[1]$ shows that order 1 is possible, and the \PMlinkescapetext{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{center} $\begin{cases} x + y + z +w &= m \\ x + y - z - w &= 0 \\ x - y + z -w &= 0 \\ x - y - z + w &= 0. \end{cases}$ \end{center} Adding the 4 equations together we get $$ 4x = m. $$ so that $m$ must be divisible by 4.