Why it matters

Matrix expo turns huge-n recurrences into fast computation. Essential technique.

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The architecture

Encode recurrence as matrix M.

State transitions: v_next = M @ v.

M^n via fast exponentiation.

Matrix exponentiationEncode Mtransition matrixFast expoM^n by squaringApply to initM^n @ v_0Fibonacci in O(log n): [[1,1],[1,0]]^n gives n-th number
Matrix expo flow.
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How it works end to end

Fibonacci matrix: [[1,1],[1,0]]. F_n from M^n.

General linear recurrence: encode all state.

Modular arithmetic for large n.