Editing Pseudo-Hadamard transform (section)

==Generalization==

The above equations can be expressed in [[Matrix (mathematics)|matrix algebra]], by considering ''a'' and ''b'' as two elements of a vector, and the transform itself as multiplication by a matrix of the form:

:<math>H_1 = \begin{bmatrix} 2 & 1 \\ 1 & 1 \end{bmatrix}</math>

The inverse can then be derived by [[Invertible matrix|inverting]] the matrix.

However, the matrix can be generalised to higher dimensions, allowing vectors of any power-of-two size to be transformed, using the following recursive rule:

:<math>H_n = \begin{bmatrix} 2 \times H_{n-1} & H_{n-1} \\ H_{n-1} & H_{n-1} \end{bmatrix}</math>

For example:

:<math>H_2 = \begin{bmatrix} 4 & 2 & 2 & 1 \\  2 & 2 & 1 & 1 \\ 2 & 1 & 2 & 1 \\ 1 & 1 & 1 & 1 \end{bmatrix}</math>