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I am trying to understand Hadamard matrix.

My question is simply about whether Hadamard matrix can always be made equivalent (by multiplying row/column by -1 or swapping rows/columns) to a symmetric one?

I only tried a couple but could not find online answers on whether symmetric Hadamard matrix of any order always exists. Thanks for the guidance!

10understanding
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1 Answers1

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Question: Let $H$ be a Hadamard matrix and $n$ be its order. Can $H$ always be made equivalent (by multiplying row/column by -1 or swapping rows/columns) to a symmetric one?

The answer can be divided into two parts.

Part 1:

For $n<16$, the answer is always YES.

First, we can eazily prove that

$$ n<16 \implies n=1,2,4,8,12 $$

Then from Reference $1$ to Reference $5$, we found that at least one symmetric Hadamard matrix $S$ exists for each order above. According to Reference $6$, we know that the total number of equivalent classes is always $1$. So given any Hadamard matrix $H$, we always have $H$ and $S$ are in the same equivalent class.

Part 2:

For $n>15$, the answer is NO.

For $n=16$, from Reference 6, we know that the total number of equivalent classes is $5$. We cannot guarantee that $H$ and $S$ are in the same equivalent class any more. In fact, by computer bruteforce search, we can see that the following matrix cannot be made equivalent to a symmetric one. (Here, $−$ stands for $−1$ and $+$ for $+1$.)

$++++++++++++++++$

$+-+-+-+-+-+-+-+-$

$++--++--++--++--$

$+--++--++--++--+$

$++++----++++----$

$+-+--+-++-+--+-+$

$++----++++----++$

$+--+-++-+--+-++-$

$++++++++--------$

$+++-+------+-+++$

$++-+---+--+-+++-$

$++---++---+++--+$

$+-++-+---+--+-++$

$+-+---++-+-+++--$

$+--++-+--++--+-+$

$+---++-+-+++--+-$

References:

  1. https://blog.csdn.net/mcs_liyun/article/details/139578602
  2. https://blog.csdn.net/mcs_liyun/article/details/139578530
  3. https://blog.csdn.net/mcs_liyun/article/details/139578492
  4. https://blog.csdn.net/mcs_liyun/article/details/139578576
  5. https://blog.csdn.net/mcs_liyun/article/details/139578443
  6. https://oeis.org/A007299
user1116616
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