Problem statement:

Show that the following recursive description provides a Hadamard matrix of each dimension $m=2^k$, $k=0,1,2,...:$

$H_0=[1]$
$H_{k+1}=\left[\begin{array}[pos]{cc}
H_k & H_k \\
H_k & -H_k \\
\end{array}\right]$

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Attempt:
The transpose of a Hadamard matrix is equal to its inverse times a constant factor.
$H_{k+1}H_{k+1}^T\left[\begin{array}[pos]{cc}
H_k & H_k \\
H_k & -H_k \\
\end{array}\right] \left[\begin{array}[pos]{cc}
H_k & H_k \\
H_k & -H_k \\
\end{array}\right]^T=2\left[\begin{array}[pos]{cc}
H_k^2 & 0 \\
0 & H_k^2 \\
\end{array}\right]$

$H_k^2=2^kI_{2^k}$

$H_{k+1}H_{k+1}^T=2^{k+1} \left[\begin{array}[pos]{cc}
I_{2^k} & 0 \\
0 & I_{2^k} \\
\end{array}\right]$

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I've not heard of Hadamard matrices before, funky stuff.
Hope someone has the time to look over my work and tell me if it's getting warmer or if it's really cold
Thanks!