The Eigenspaces of the fast Walsh Hadamard transform

 




For the normalized Hadamard transform H (orthogonal, symmetric, leaves vector magnitude unchanged and self-inverse), you can say:

It has exactly two eigenspaces: one for eigenvalue +1 and one for eigenvalue −1.

Given a vector x: 

u =(I+H)x

v = (I-H)x

(I+H)u = u

(I-H)v = -v


x = 0.5(u+v)

Hx = 0.5(u-v)

That’s actually quite special, and it connects to several interesting areas in mathematics:

  • Two-point spectrum: rare for nontrivial transforms.

  • Perfect involution: exactly, no scaling.

  • Orthogonal split of the entire space into just two halves, making projection filtering trivial.

  • Group-theoretic structure: eigenspaces correspond to symmetry classes of the Boolean cube.

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