Editing Hadamard matrix (section)

===Proof===
The [[mathematical proof|proof]] of the nonexistence of Hadamard matrices with dimensions other than 1, 2, or a multiple of 4 follows:

If <math>n>1</math>, then there is at least one scalar product of 2 rows which has to be 0. The scalar product is a sum of ''n'' values each of which is either 1 or −1, therefore the sum is [[parity (mathematics)|odd]] for odd ''n'', so ''n'' must be [[parity (mathematics)|even]].

If <math>n = 4 m + 2</math> with <math>m \geq 1</math>, and there exists an <math>n \times n</math> Hadamard matrix <math>H = (h_{i,j})_{i,j \in \{0,1,...,n-1\}}</math>, then it has the property that for any <math>k \neq l</math>:
:<math>\sum_{i=0}^{n-1} h_{k,i} h_{l,i} = 0</math>

Now we define the matrix <math>A = (a_{i,j})_{i,j \in \{0,1,...,n-1\}}</math> by setting <math>a_{i,j} = h_{0,j}h_{i,j}</math>.
Note that <math>A</math> has all 1s in row 0.
We check that <math>A</math> is also a Hadamard matrix:
:<math>\sum_{i=0}^{n-1} a_{k,i} a_{l,i} = \sum_{i=0}^{n-1} h_{0,j} h_{k,i} h_{0,j} h_{l,i} = \sum_{i=0}^{n-1} h_{0,j}^2 h_{k,i} h_{l,i} = \sum_{i=0}^{n-1} h_{k,i} h_{l,i} = 0.</math>

Row 1 and row 2, like all other rows except row 0, must have <math>n/2</math> entries of 1 and <math>n/2</math> entries of −1 each. (*)

Let <math>\alpha</math> denote the number of 1s of row 2 beneath 1s in row 1.
Let <math>\beta</math> denote the number of −1s of row 2 beneath 1s in row 1.
Let <math>\gamma</math> denote the number of 1s of row 2 beneath −1s in row 1.
Let <math>\delta</math> denote the number of −1s of row 2 beneath −1s in row 1.

Row 2 has to be orthogonal to row 1, so the number of products of entries of the rows resulting in 1, <math>\alpha + \delta</math>, has to match those resulting in −1, <math>\beta + \gamma</math>.
Due to (*), we also have <math>n/2 = \alpha + \gamma = \beta + \delta</math>, from which we can express <math>\gamma = n/2 - \alpha</math> and <math>\delta = n/2 - \beta</math> and substitute:

:<math>\alpha + \delta = \beta + \gamma</math>
:<math>\alpha + \frac{n}{2} - \beta = \beta + \frac{n}{2} - \alpha</math>
:<math>\alpha - \beta = \beta - \alpha</math>
:<math>\alpha = \beta</math>

But we have as the number of 1s in row 1 the odd number <math>n/2 = \alpha + \beta</math>, [[proof by contradiction|contradiction]].