10 Dimensional Matrix Power Calculator

Matrix Power Calculation:

\[ A^k = A \times A \times \dots \times A \text{ (k times)} \]

or using diagonalization when possible

Matrix A (10x10):

Exponent (k):

unitless

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1. What is Matrix Power?

Matrix power A^k is the matrix A multiplied by itself k times. It's fundamental in linear algebra and has applications in computer graphics, physics, and Markov chains.

2. How Matrix Power is Calculated

There are two main methods:

Direct multiplication: \( A^k = A \times A \times \dots \times A \) (k times)

Diagonalization: \( A = PDP^{-1} \) then \( A^k = PD^kP^{-1} \)

This calculator uses direct multiplication which is more universally applicable.

3. Applications of Matrix Powers

Details: Matrix powers are used in:

Graph theory (counting paths between nodes)
Physics (quantum mechanics, dynamical systems)
Computer science (graphics transformations, algorithms)
Economics (input-output models)

4. Using the Calculator

Tips:

Enter all 100 elements of the 10×10 matrix
The exponent must be a positive integer
For large exponents (>10), calculation may take longer

5. Frequently Asked Questions (FAQ)

Q1: What's the computational complexity?
A: O(n³) per multiplication where n is matrix dimension (here n=10).

Q2: Can I use non-integer exponents?
A: Not with this calculator. Non-integer exponents require matrix diagonalization.

Q3: What's the largest exponent I can use?
A: Technically unlimited, but practical limits depend on your computer.

Q4: Why does my matrix show all zeros?
A: You might have entered a nilpotent matrix or the result is very small.

Q5: How accurate are the results?
A: Results are limited by PHP's floating-point precision (about 14 decimal digits).