The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on 2m real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size 2 × 2 × ⋯ × 2 × 2. It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (adamaʁ), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. The Hadamard transform Hm is a 2m × 2m matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2m real numbers xn into 2m real numbers Xk. The Hadamard transform can be defined in two ways: recursively, or by using the binary (base-2) representation of the indices n and k. Recursively, we define the 1 × 1 Hadamard transform H0 by the identity H0 = 1, and then define Hm for m > 0 by: where the 1/ is a normalization that is sometimes omitted. For m > 1, we can also define Hm by: where represents the Kronecker product. Thus, other than this normalization factor, the Hadamard matrices are made up entirely of 1 and −1. Equivalently, we can define the Hadamard matrix by its (k, n)-th entry by writing where the kj and nj are the bit elements (0 or 1) of k and n, respectively. Note that for the element in the top left corner, we define: . In this case, we have: This is exactly the multidimensional DFT, normalized to be unitary, if the inputs and outputs are regarded as multidimensional arrays indexed by the nj and kj, respectively. Some examples of the Hadamard matrices follow. where is the bitwise dot product of the binary representations of the numbers i and j.