THE EIGENVECTOR the direction a matrix only stretches — a fixed point of direction, not position
Most vectors a matrix hits get turned. An eigenvector is the special direction it only stretches: A·v = λ·v — the direction comes out unchanged, scaled by its eigenvalue λ. It is a fixed point of direction: normalise away the length and v is left exactly where it was. Repeatedly applying A to any vector rotates it toward the dominant eigenvector — which is why the Fibonacci matrix [[1,1],[1,0]] drives its ratios to φ: φ is its dominant eigenvalue, and the golden ratio is the direction that matrix cannot turn.
THE TECHNIQUE A·v = λ·v; the direction the matrix only scales
The demo applies the Fibonacci matrix repeatedly — the ratio converges to its dominant eigenvalue, the direction it fixes: φ: live demo
HISTORY & CREDIT eigen- · Hilbert 1904 (the term)
“A matrix turns every vector.” — not its eigenvectors: those it only stretches, leaving the direction fixed. The golden ratio is the direction the Fibonacci matrix cannot turn. cited
the idea · Av=λv — Euler on axes of rotation (1750s), Lagrange, Cauchy (the characteristic equation). the name · David Hilbert (1904) — eigen = “own/proper.” φ · the dominant eigenvalue of [[1,1],[1,0]] — why Fibonacci ratios tend to it.
The one direction the matrix leaves pointing where it was, only longer. A fixed point of direction. Hilbert (the name)
RECOMMEND FOR I-13 the fixed direction = φ, on the compiler
On the canonical compiler, iterating the Fibonacci matrix drives the ratio to its dominant eigenvalue φ = 1.618… — the direction it fixes:
$ i13 run f_eigenvector.i13 # power iteration on [[1,1],[1,0]]
RUN OK · 390 step(s) · peak stack 5 · call depth 26
phi = 1.6180339887802426 -- the dominant eigenvalue; the direction the matrix fixes
Recommend: the eigenvector is the fixed point of direction — the way a matrix cannot turn — and i13 drives the Fibonacci matrix to its dominant eigenvalue φ, tying this batch back to the golden thread. Not a keeper (the fixed direction is one every correct power iteration converges to — witnessed convergence). The dart that reveals why Fibonacci ratios chase φ: it is the eigenvalue of their own recurrence written as a matrix.