1. What is Matrix Power?

Matrix power (A^k) is the matrix multiplied by itself k times. For diagonalizable matrices, it can be computed efficiently using eigenvalues and eigenvectors.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( A \) — Square matrix
• \( k \) — Exponent (integer)
• \( P \) — Matrix of eigenvectors
• \( D \) — Diagonal matrix of eigenvalues
• \( P^{-1} \) — Inverse of P

Explanation: The equation allows efficient computation of matrix powers by working with the diagonalized form.

3. Importance of Matrix Powers

Details: Matrix powers are fundamental in many applications including Markov chains, differential equations, and computer graphics transformations.

4. Using the Calculator

Tips: Enter the matrix in format "1,2;3,4" for [[1,2],[3,4]]. The matrix must be square. Enter integer exponent.

5. Frequently Asked Questions (FAQ)

Q1: What matrices are diagonalizable?
A: A matrix is diagonalizable if it has n linearly independent eigenvectors (n = matrix size).

Q2: What if my matrix isn't diagonalizable?
A: For non-diagonalizable matrices, other methods like Jordan form or direct multiplication must be used.

Q3: Can I use fractional exponents?
A: This calculator only handles integer exponents. Fractional exponents would require matrix logarithm/exponential.

Q4: What's the computational complexity?
A: Diagonalization is O(n³) but then powers are O(n²), much better than O(kn³) for direct multiplication.

Q5: How accurate are the results?
A: Accuracy depends on eigenvalue computation. Ill-conditioned matrices may give inaccurate results.