Suppose A is a Hadamard matrix of size $d$. Let A be in normalized in a sense that first row and first column are all ones. What is the sum of rows? I tried random Hadamard matrices and seem to get $d,0,0,0,0,\ldots$


Follows immediately from the property of Hadamard Matrices: Any two distinct rows are orthogonal.

Since all the other rows are orthogonal to the all ones row, the sum of the elements in each of those rows must be zero.

EDIT: To answer Qiaochu's query:

MUltiplying a row or column of a Hadamard matrix by -1 gives yet another Hadamard matrix.

Using this operation, we can easily normalize a Hadamard matrix to make the elements of the first row to be all ones.

  • $\begingroup$ Again, I do not see why an all-ones row need exist. If this is a known theorem about Hadamard matrices, do you have a citation? It does not appear to be stated anywhere in the Wikipedia article. $\endgroup$ Sep 16 '10 at 17:32
  • 2
    $\begingroup$ @Qiaochu: The fact that the first row is all ones is an assumption of the problem statement! No one is claiming that all Hadamard matrices have that property. $\endgroup$
    – Aryabhata
    Sep 16 '10 at 17:34
  • $\begingroup$ @QIaochu: btw, multiplying a row/column by -1 gives another hadamard matrix, so we can normalize it. So I guess that answers your query. $\endgroup$
    – Aryabhata
    Sep 16 '10 at 17:49

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