Generate a Walsh Matrix

MATL, 4 bytes

W4YL

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How it works:

W       % Push 2 raised to (implicit) input
4YL     % (Walsh-)Hadamard matrix of that size. Display (implicit)

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Without the built-in: 11 bytes

1i:"th1M_hv

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How it works:

For each Walsh matrix W, the next matrix is computed as [W W; W −W], as is described in the challenge. The code does that n times, starting from the 1×1 matrix [1].

1       % Push 1. This is equivalent to the 1×1 matrix [1]
i:"     % Input n. Do the following n times
  t     %   Duplicate
  h     %   Concatenate horizontally
  1M    %   Push the inputs of the latest function call
  _     %   Negate
  h     %   Concatenate horizontally
  v     %   Concatenate vertically
        % End (implicit). Display (implicit)

Perl 6, 63 44 40 bytes

{map {:3(.base(2))%2},[X+&] ^2**$_ xx 2}

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Non-recursive approach, exploiting the fact that the value at coordinates x,y is (-1)**popcount(x&y). Returns a flattened array of Booleans.

-4 bytes thanks to xnor's bit parity trick.

Haskell, 57 56 bytes

(iterate(\m->zipWith(++)(m++m)$m++(map(0-)<$>m))[[1]]!!)

Try it online! This implements the given recursive construction.

-1 byte thanks to Ørjan Johansen!