Constructing symmetric nonnegative matrices via the fast Fourier transform (Q1827252)

Given a set of real numbers \(\sigma = \{\lambda_1, \lambda_2, \dots, \lambda_n\}\), the authors propose a new method for constructing a real \(n\times n\) symmetric matrix \(S\) with eigenvalues \(\sigma\). The proposed method proceeds by constructing a real and symmetric circulant matrix \(C\) of double dimension so that its (double) eigenvalues are given by \(\tilde \sigma = \{\mu, \lambda_2, \dots, \lambda_n\}\), where \(\mu = - (\lambda_2 + \cdots + \lambda_n)\). Then, after a simple block diagonalization of \(C\), a symmetric \(n \times n\) matrix \(\tilde S\) having eigenvalues \(\tilde \sigma\) is obtained as one of the diagonal blocks. Finally, the matrix \(S\) is constructed from a certain rank-one correction of \(\tilde S\). In the second part of the paper, the authors prove several sufficient conditions under which their method produces an irreducible nonnegative matrix \(S\).