We propose a novel method to embed a functional magnetic resonance imaging (fMRI) dataset in a low-dimensional space. The embedding optimally preserves the local functional coupling between fMRI time series and provides a low-dimensional coordinate system for detecting activated voxels. To compute the embedding, we build a graph of functionally connected voxels. We use the commute time, instead...

Abstract As an extension of Sylvester’s matrix, a tridiagonal matrix is investigated by determining both left and right eigenvectors. Orthogonality relations between eigenvectors are derived. Two determinants the matrices constructed evaluated in closed form.

The partition function for the problem of an arbitrary number of directed nonintersecting walks interacting with one or two walls parallel to the direction of the walks is calculated exactly utilizing a theorem recently proved concerning the Bethe ansatz for the eigenvectors of the transfer matrix of the five-vertex model. This theorem shows that the completeness of the Bethe ansatz eigenvector...

The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant m...

Emergence of localized eigenvectors can cause a failure of standard spectral method for graph partitioning. The spectral method using the non-backtracking matrix was proposed as a way to overcome this problem. The numerical experiments on several examples of real networks show that, indeed, the non-backtracking matrix does not exhibit localization of eigenvectors. We show that, however, localiz...