Hadamard matrix: n×n matrix of ±1 with H·H^T=nI. Sylvester (1867): H_1=[1], H_{2n}=H_n⊗[[1,1],[1,-1]]. Exists for n=1,2,4,8,16,... Hadamard conjecture: exists for all n≡0(mod4). The largest unresolved order is n=668.
How to Use
Enter k for n=2^k:
- Matrix: H_n display
- Verify: H·H^T=nI
- Determinant: Maximal
Applications
Walsh-Hadamard transform (fast spectral analysis), CDMA telecommunications (spreading codes), quantum computing (Hadamard gate), compressed sensing, and design of experiments. The Hadamard transform is O(n log n) like FFT.
The Conjecture
Hadamard (1893) conjectured: Hadamard matrices exist for all orders n≡0(mod4). Verified for n≤668 (with some gaps). The Paley construction uses quadratic residues for n=p+1 (p prime). Many constructions exist but no universal method.
Step-by-Step Instructions
- 1Enter k.
- 2Generate H_{2^k}.
- 3Verify orthogonality.
- 4Check determinant.
- 5See Walsh relation.