On simple bounds for eigenvalues of symmetric tridiagonal matrices

How much can be said about the location of the eigenvalues of a symmetric tridiagonal matrix just by looking at its diagonal entries? We use classical results on the eigenvalues of symmetric matrices to show that the diagonal entries are bounds for some of the eigenvalues regardless of the size of the off-diagonal entries. Numerical examples are given to illustrate that our arithmetic-free technique delivers useful information on the location of the eigenvalues.

The bounds of education in the human brain connectome

Inter-individual heterogeneity is evident in aging; education level is known to contribute for this heterogeneity. Using a cross-sectional study design and network inference applied to resting-state fMRI data, we show that aging was associated with decreased functional connectivity in a large cortical network. On the other hand, education level, as measured by years of formal education, produced an opposite effect on the long-term. These results demonstrate the increased brain efficiency in individuals with higher education level that may mitigate the impact of age on brain functional connectivity.