Why eigenvectors matter

The first launch reads ∠ to u1 $= 63.00°$ and rayleigh q $= 1.412$. One press: $33.19°$ and $2.401$. Then $12.30°$ and $2.909$, then $4.16°$ and $2.989$ — once small, each press cuts the angle to roughly a third. By the fifth press the quotient reads $3.000$; every restart funnels onto the same dashed ray.

Write the start as a mix of the eigendirections, $\mathbf{x}_0 = c_1\mathbf{v}_1 + c_2\mathbf{v}_2$. Each press scales each piece by its own eigenvalue — the pieces never interfere:

The $\mathbf{v}_1$ share compounds while the $\mathbf{v}_2$ share stalls, so their ratio is $3^k$ and the direction swings onto the dominant ray — unless the start has $c_1 = 0$, exactly on the $\lambda_2$ line. Normalizing never touches direction — this loop is power iteration. The Rayleigh quotient $\mathbf{v}\cdot A\mathbf{v} \,/\, \mathbf{v}\cdot\mathbf{v}$ reads the eigenvalue of the current direction — hence the lock onto $3.000$.

PageRank is this loop at web scale. Rank a page by the rank of the pages linking to it: that circular definition is an eigenvector equation for the web's link matrix, and Google's original ranking is its dominant eigenvector — found by power iteration: the loop only multiplies and renormalizes, even at billions of pages.

Principal component analysis plays the same card on data. The eigenvectors of a covariance matrix are the natural axes of a point cloud, each eigenvalue measuring the spread along its axis; keeping the few largest-eigenvalue directions compresses the data while preserving most of its structure.