Exercise 2.7

Question

A Hadamard matrix is a matrix whose entries are all ±1 and whose
transpose is equal to its inverse times a constant factor. It is known that if
$A$ is a Hadamard matrix of dimension $m > 2$, then m is a multiple of $4$. It is
not known, however, whether there is a Hadamard matrix for every such $m$,
though examples have been found for all cases $m \leq 424$.
ewline

Show that the following recursive description provides a Hadamard matrix of
each dimension $m = 2^k$, $k = 0, 1,2, \ldots$:
$$H_0 = \begin{bmatrix} 1 \end{bmatrix} \qquad H_{k+1} = \begin{bmatrix} H_k & H_k \ H_k & -H_k \end{bmatrix}.$$

Desh Raj · Accepted Answer

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Exercise 2.7

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Exercise 2.7

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