Problem $8\mathrm{.}4($Orthogonal basis made of coordinates $+-1).$ A Hadamard matrix is a matrix $H$ whose entries are all $+-1,$ and the columns are mutually orthogonal $($but they don't have to be unit vectors$).($When $n$ is a power of $2,$ then an example of Hadamard matrix would be the matrix made by the Haar wavelet basis.$)$

For any $nn$ Hadamard matrix, compute $H^{T}H.$

If $H$ is a Hadamard matrix, show that $\left[\begin{array}{cc}H& H\\ H& -H\end{array}\right]$ is also a Hadamard matrix.

Can you find a $33$ Hadamard matrix? Find it or show why not.

Let $H$ be any Hadamard matrix. Then if we permute the rows and columns, or if we negate some rows and columns $($i$.$e$.,$ multiply the row or culumn by $-1),$ prove that the result is still a Hadamard matrix. $($If two Hadamard matrices can be obtained from each other like this, then we say they are equivalent.$)$

Prove that all $44$ Hadamard matrices are equivalent.

$($Read only$)$ It is conjectured that a Hadamard matrix should exist for all $(4k)(4k)$ matrices. However, this remains unproven to this date. For examlpe, is there a $668668$ Hadamard matrix? We don't know the answer yet, as far as I can tell. The Haar wavelet basis gives a Hadamard matrix whenever $n$ is a power of $2.$ But, for examlpe, try to find a Hadamard matrix when $n=12,$ if you want a challenge. And on the question of equivalence, in general, Hadamard matrices of the same size could be inequivalent. For example, there are five inequivalent $1616$ Hadamard matrices.