14.8a) indicated that 63% of the among-cell variation (Ra2) of the community composition (159 species) was spatially structured and explained by the 339 spatial eigenfunctions. Nearly half of that (0.278/0.626 = 44%) was also explained by the four topographic variables. Without surprise, ...
Finally, we have that minimum value of the Hamiltonian subject to the spherical constraint is the minimum eigenvalue of the repelling Laplacian. This minimum is a global minimum for the system defined in Eq. (3) with the spherical constraint, and the associated one-dimensional embedding is ...
gradient flow, SIAM J. Sci. Comput. 25 (2004) 1674-1697] or the damped inverse iteration suggested in [P. Henning and D. Peterseim, Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: Global convergence and computational efficiency, SIAM J. Numer. Anal. 58 (2020) 1744-1772...
The complete graph Kn has a simple eigenvalue n − 1 and eigenvalue − 1 of multiplicity n − 1. Hence, PKn(λ)=(λ−n+1) (λ+1)n−1; • The complete bipartite graph Kn1,n2 has two simple eigenvalues ±n1n2 and eigenvalue 0 of multiplicity n − 2. Hence, PKn1...
Let be an eigenvalue of having algebraic multiplicity equal to . Let be the generalized eigenspace associated to . Then, the dimension of is . ProofSolved exercisesBelow you can find some exercises with explained solutions. Exercise 1In an example above we have found two generalized eigenvectors ...
reproduce the original spectra. For the group of similar spectra used to generate the covariance matrix, the importance of each eigenvector is related to its eigenvalue. The ratio of an eigenvalue to the sum of the eigenvalues gives the amount of variation explained by the associated eigen...
and the corresponding eigenvalue as λ full =λ ′ 1 E −1 +· ·· . (3) In (2) we have constrained |1 full to have unit norm. From the eigenvalue equation H|1 full = λ full |1 full we find at lowest order c ′ A j = − A j |H|1 λ A j . (4) 2 We...
The principal eigenvector, linked to the maximum eigenvalue, signifies the predominant direction of phase synchronization among brain regions at that particular moment. To discern recurring connectivity states, the leading eigenvectors were subsequently categorized using the k-means clustering algorithm, ...
The function relation has been detected between MC value for a mapped eigenvector and its corresponding eigenvalue as follows: MCj = n 1TC1 λj (3) Based upon these properties, the eigenvectors can be interpreted as follows: The first eigenvector E1 is the set of real numbers that has ...