Matrix Exponentiation: Mental Model

Start with the idea

Begin with the concrete question: Return Fibonacci F(50). For the visible input n=50, the stated result is 12586269025. Before reading code, name the smallest piece of information that must survive from one step to the next. An arithmetic or bitwise transformation relies on an identity. Write that identity and the domain in which it holds. That is the mental model for Matrix Exponentiation; it gives you a reason for each update instead of a line of code to memorise.

The core mechanism is this: Encode a linear recurrence as a matrix and exponentiate the transition by squaring. Treat it as a claim to test after every operation. Write down the state before the first operation, update it once, and ask which part of the claim is still true. If an item is skipped, inserted, or removed, explain why that action cannot destroy an answer that should be retained. A useful trace has columns for the current input position, the state before the step, the action, and the state afterward. Use the visible sample until the expected result becomes inevitable rather than merely plausible.

Now inspect the Rust sketch, not just its final expression. The supplied Rust sketch is a starting point.