Spectral properties of the binary Hadamard matrices

ABSTRACT

A binary Hadamard matrix (also called an S-matrix) is an $n\times n$ $(0,1)$-matrix formed by taking an $(n+1)\times (n+1)$ Hadamard matrix in which the entries in the first row and column are $+1$, changing $+1$'s to 0's and $-1$'s to $+1$'s, and deleting the first row and column. In this paper, some spectral properties of the binary Hadamard matrices are derived. All singular values and eigenvalues' modulus of an arbitrary S-matrix are obtained. Two special types of binary Hadamard matrices, namely the symmetric and the skew-type ones, are analysed in more detail. In particular, we prove that an S-matrix $S_n$ (regardless of its order n) is of skew type if, and only if, all its eigenvalues different from the largest one (in modulus) are imaginary and have real part 1/2. Finally, the symmetric and skew-symmetric parts of an S-matrix are analysed.

KEYWORDS:

• Hadamard matrix
• S-matrix
• symmetric S-matrix
• skew-type S-matrix
• spectral properties
• Frobenius norm

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 15A18 Eigenvalues, singular values, and eigenvectors
• 15B34 Boolean and Hadamard matrices

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work has been partially supported by the ‘Ministerio de Economía y Competitividad’ (Spanish Government), and FEDER [grant contract number CTM2014-55014-C3-1-R].