Definition

A Hadamard matrix is a square matrix that has entries of −1-1 or +1+1, with mutually orthogonal rows.

Sylvester's construction

For order nn Hadamard matrix, may construct order 2n2n Hadamard matrix by

[HHH−H]\begin{bmatrix} H & H \\ H & -H \end{bmatrix}

Notes

References

  1. https://en.wikipedia.org/wiki/Hadamard_matrix
  2. https://mathworld.wolfram.com/HadamardMatrix.html
  3. Sylvester, J. J. L. X. Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton’s Rule, ornamental tile-work, and the theory of numbers. Philos. Mag. 34, 461–475 (1867). https://doi.org/10.1080/14786446708639914
  4. Suksmono, A.B., Minato, Y. Quantum computing formulation of some classical Hadamard matrix searching methods and its implementation on a quantum computer. Sci Rep 12, 197 (2022). https://doi.org/10.1038/s41598-021-03586-0