Matrix Power Calculator
Calculate powers of a square matrix using efficient algorithms.
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Matrix Powers
Matrix power \(A^n\) is defined as multiplying matrix A by itself n times:
\[ A^n = \underbrace{A \times A \times \cdots \times A}_{n \text{ times}} \]
Special cases:
- \(A^0 = I\) (Identity matrix)
- \(A^1 = A\)
- \(A^2 = A \times A\)
Properties:
- \(A^m \times A^n = A^{m+n}\)
- \((A^m)^n = A^{mn}\)
- \((AB)^n \neq A^nB^n\) in general
- If A is invertible, \(A^{-n} = (A^{-1})^n\)
Applications:
- Markov chains and probability
- Graph theory (adjacency matrices)
- Solving systems of differential equations
- Population dynamics
- Computer graphics transformations
Efficient Computation:
- Binary exponentiation for large powers
- Diagonalization for special matrices
- Jordan form for defective matrices