## You are here

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

## Primary tabs

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

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.

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

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.

Major Section:

Reference

Type of Math Object:

Proof

Parent:

## Mathematics Subject Classification

05B20*no label found*15-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections