I suggest you to take a look:

P. Embury and A. Rao, A Path To Hadamard Matrices, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes Lecture Notes in Computer Science Volume 851, 2007, pp 281-290.

          There are characteristics of Hadamard matrices that enable an exhaustive search using algorithmic techniques. The search derives primarily from the eigenvalues which are constant after the Hadamard matrix is multiplied by its transpose. Generally this would be a performance concern but there are additional properties that enable the eigenvalues to be predicted. Here an algorithm is given to obtain a Hadamard matrix from a matrix of 1s using optimisation techniques on a row-by-row basis.

Please see the attachment in the following useful link:

https://researchbank.rmit.edu.au/view/rmit:1164/n2006006652.pdf

Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that every two different rows in a Hadamard matrix represent two perpendicular vectors, while in combinatorial terms, it means that every two different rows have matching entries in exactly half of their columns and mismatched entries in the remaining columns

Have you try with the Hadamard product for reciprocal matrices?

Thank you very much for your answers. They are very useful to me.