2026-02-12 | Understand Rayleigh Quotient for Eigenvectors

Goal: Understand Rayleigh Quotient for Eigenvectors

Summary: Previous days did not do documentation, will be more dilligent about documenting! Focusing on first principles intuitions for deriving eigenvalues as maximal disharmony of a graph

Work sessions

In	Out
14:40	15:10
20:00	20:30
22:30	22:55

Super interesting! I didn't fully intuit the \(f^\top L f = \sum_{(i,j) \in E} (f_i - f_j)^2\) but understood that the Rayleigh Quotient for a matrix \(M\) returns the eigenvalue when given the eigenvector as input.

\[R(v ; M) = \lambda\]

where \(v\) is an eigenvector of \(M\) and \(\lambda\) is the corresponding eigenvalue.

Next step tomorrow is to fully gain intuition on the meaning of the Rayleigh Quotient applied to the Laplacian matrix.