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A lower bound for the nearest correlation matrix problem based on the circulant mean. (English) Zbl 1316.15024

This paper finds a lower bound for the distance of a given symmetric matrix \(A\) from the set of correlation matrices, i.e., those semidefinite matrices whose diagonal entries all equal 1. As the distance between \(A_c\) and its nearest correlation matrix can be derived from the eigenvalues of \(A_c\), the lower distance bound is taken here as the distance from \(A\) to its circulant mean \(A_c\). This bound is then applied to find the asymptotic behavior of the distances of \(n\) by \(n\) Laplacian matrices with Dirichlet boundary conditions from the set of correlation matrices.

MSC:

15A45 Miscellaneous inequalities involving matrices
15B05 Toeplitz, Cauchy, and related matrices
90C22 Semidefinite programming
15A18 Eigenvalues, singular values, and eigenvectors
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References:

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