Math Explained to Programmers: The Fibonacci Sequence- Using State Matrix Approach

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The story

Today, let’s dive into a fresh way to think about the Fibonacci sequence: using matrices and eigenvalues.

Let math do the heavy lifting this time. explained step by step.

Fibonacci Sequence

The Fibonacci sequence is defined as:

F(n) = F(n-1) + F(n-2)

with initial conditions:

F(0) = 0, F(1) = 1

1. Define State Vector

To represent the Fibonacci sequence in matrix form, define a state vector:

vₙ = [F(n)    ]
     [F(n-1)]

What does this mean? Fibonacci numbers can be represented as vectors, enabling them to be transformed by a matrix like this:

vₙ = A · vₙ₋₁, where A = [1 1]
                        [1 0]

What is the role of matrix A?

Matrix A is the state transfer matrix, used to transform the vector vₙ₋₁ into vₙ. Think of A as a function traversing the Fibonacci sequence step by step.

The initial condition is:

v₁ = [F(1)] = [1]
     [F(0)]   [0]

Step-by-step progression:

v₂ = A · v₁ =  [1 1] · [1] = [1]
               [1 0]…