This question is from The Theory of Error-Correcting Codes by MacWilliams and Sloane, Problem 2.(3).
If n = $2^m$, let $u_1$, $u_2$,...,$u_n$ denote the distinct $m$-tuples. Show that the matrix $H = (h_{ij})$ where ($h_{ij}$) = $(-1)^{u_i \cdot u_j}$ is a Hadamard matrix and is equivalent to Sylvester-type Hadamard matrix, where Sylvester-type Hadamard matrix is of form $H_1 = (1)$, and $H_n = \left( \begin{array}{ccc} H_{n/2} & H_{n/2} \\ H_{n/2} & -H_{n/2} \\ \end{array} \right) $.
I had already proved that $H$ is a Hadamard matix, since we can consider two rows of $H$ and their dot product, we have $[(-1)^{u_{1} \cdot u_{j}},...,(-1)^{u_{n} \cdot u_{j}}] \cdot [(-1)^{u_{1} \cdot u_{k}},...,(-1)^{u_{n} \cdot u_{k}}] = (-1)^{u_{1} \cdot (u_{j}+u_{k})}+...+(-1)^{u_{n} \cdot (u_{j}+u_{k})}$, let $u_{j}+u_{k}$ has $d$ $1$s and $m-d$ $0$s, $2^{m-d} \cdot \sum\limits_{i \text{ even}} \left( \begin{array}{ccc} d \\ i \\ \end{array} \right) =2^{m-1} = n/2$. So half terms in $(-1)^{u_{1} \cdot (u_{j}+u_{k})}+...+(-1)^{u_{n} \cdot (u_{j}+u_{k})}$of will be $1$, others will be $-1$. Thus $[(-1)^{u_{1} \cdot u_{j}},...,(-1)^{u_{n} \cdot u_{j}}] \cdot [(-1)^{u_{1} \cdot u_{k}},...,(-1)^{u_{n} \cdot u_{k}}] = 0$ if $j \neq k$ and $=n$ if $j = k$ since $H = H^{tr}$. But I don't know how to prove that $H$ is equivalent to Sylvester-type Hadamard matrix, could anyone give me some comments or tips? Thanks a lot.