What diagonalisation is for

Why go to all this trouble? Because in an eigenbasis, hard computations become multiplications of numbers.

Powers of a matrix

This is the flagship application. Writing A = P D P⁻¹:

A² = (P D P⁻¹)(P D P⁻¹) = P D (P⁻¹P) D P⁻¹ = P D² P⁻¹

and by induction:

A^k = P D^k P⁻¹        with        D^k = [ λ1^k   0   ]
                                         [  0   λ2^k  ]

Raising a diagonal matrix to the power k means raising each entry to the power k. Computing A^100, out of reach by repeated multiplication by hand, becomes immediate.

Recurrence sequences

A sequence defined by a linear recurrence can be put in matrix form. For Fibonacci u(n+1) = u(n) + u(n-1):

[ u(n+1) ]   [ 1  1 ] [ u(n)   ]              [ 1  1 ]
[ u(n)   ] = [ 1  0 ] [ u(n-1) ]        A =   [ 1  0 ]

The general term then follows by diagonalising A. Its eigenvalues are the roots of λ² - λ - 1 = 0:

λ1 = (1 + √5)/2 ≈ 1.618      (the golden ratio)
λ2 = (1 - √5)/2 ≈ -0.618

hence Binet's formula, which gives u(n) in terms of n without computing all the previous terms. And it shows at once why the ratio of consecutive terms tends to the golden ratio: λ1 having the larger modulus, it eventually dominates.

Long-term behaviour

This is the most useful reading, and it often needs no full computation:

|λ| > 1   ->  the component EXPLODES
|λ| = 1   ->  it stays STABLE
|λ| < 1   ->  it DIES OUT

The largest eigenvalue in modulus dictates the asymptotic behaviour.

This is what decides the stability of a dynamical system, the convergence of an iterative algorithm, or the damping of a structure.

Markov chains

A probability transition matrix always has eigenvalue 1. The associated eigenvector is the stationary state — the distribution the system converges to, whatever the starting state.

state(n) = M^n × state(0)   ---->   eigenvector of eigenvalue 1

This is the principle behind Google's PageRank: ranking web pages amounts to computing the dominant eigenvector of a gigantic link matrix.

Elsewhere, the same idea

Differential equations :  X' = A X decouples into n scalar equations
Mechanics              :  the normal modes of vibration of a structure
Statistics (PCA)       :  the axes of greatest variance in the data
Quantum physics        :  energy levels are eigenvalues

In all these fields, diagonalising means the same thing: finding the coordinates in which the problem splits into independent pieces.

Summary

• A^k = P D^k P⁻¹: powers become immediate.
• Linear recurrence sequences are solved by diagonalising their matrix (Fibonacci → golden ratio).
• The modulus of the largest eigenvalue dictates long-term behaviour: explosion, stability or extinction.
• A Markov chain converges to the eigenvector of eigenvalue 1 (the PageRank principle).
• The same idea appears in differential equations, vibration mechanics, PCA and quantum physics.