Algebra and abstract structures · Maths EE idea · Solid

Fibonacci numbers by matrices and eigenvalues

A research question to start from

How do eigenvalues of a 2×2 matrix give closed forms for linear recurrences such as the Fibonacci sequence, and when does the method break down?

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Why it works as a maths EE

Connects sequences, matrices and the golden ratio, with a clear failure case (repeated eigenvalues) to discuss.

Mathematics you would need

Matrix multiplication and powers
Eigenvalues and eigenvectors
Diagonalisation
Linear recurrences

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Write the Fibonacci recurrence as a matrix power.
Diagonalise and derive Binet's formula.
Generalise to other second-order recurrences.
Handle the repeated-eigenvalue case and explain what changes.

Scope and difficulty

Solid. Solid; well suited to students who enjoy algebra.

Pitfalls

Only the Fibonacci case.
Skipping the repeated-root case.

Where to start reading

Search a library catalogue or a university's open lecture notes for: Binet formula diagonalisation; linear recurrence eigenvalues repeated root. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

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