Definition

For a square matrix A and positive integer n: A^n = A × A × ... × A (n times)

Special Cases

• A^0 = I (identity matrix)
• A^1 = A
• A^2 = A × A
• A^3 = A × A × A

Properties

• A^m × A^n = A^(m+n)
• (A^m)^n = A^(m×n)
• (AB)^n ≠ A^n × B^n (in general)
• For diagonalizable matrices: A^n = P × D^n × P^(-1)

Fast Exponentiation

For large powers, use the fast exponentiation algorithm:

• If n is even: A^n = (A^(n/2))^2
• If n is odd: A^n = A × A^(n-1)
• This reduces complexity from O(n) to O(log n)

Applications

• Markov chains: Transition probabilities
• Graph theory: Path counting
• Dynamical systems: State evolution
• Linear recurrence: Fibonacci sequences