1. What is an Eigenvector Calculator?

Definition: This calculator solves for the eigenvector (v) of a square matrix A given an eigenvalue λ.

Purpose: Eigenvectors are fundamental in linear algebra, used in stability analysis, quantum mechanics, and machine learning.

2. How Does the Calculator Work?

The calculator uses the equation:

Where:

• \( A \) — Square matrix
• \( \lambda \) — Eigenvalue (scalar)
• \( I \) — Identity matrix
• \( \vec{v} \) — Eigenvector (non-zero solution)

Explanation: The calculator finds non-zero vectors that satisfy this equation, which remain in the same direction when transformed by A.

3. Importance of Eigenvectors

Details: Eigenvectors reveal the "axes" that are preserved under linear transformation, crucial for understanding system behaviors.

4. Using the Calculator

Tips:

• Enter matrix rows separated by commas, columns by spaces
• Matrix must be square (same number of rows and columns)
• Enter a known eigenvalue (use our eigenvalue calculator if needed)

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix isn't square?
A: Only square matrices have eigenvectors. The calculator will show an error for non-square matrices.

Q2: Can I get multiple eigenvectors?
A: For repeated eigenvalues, there may be multiple linearly independent eigenvectors.

Q3: Why is the eigenvector not unique?
A: Any scalar multiple of an eigenvector is also an eigenvector. We typically normalize them.

Q4: What if (A - λI) is invertible?
A: Then λ isn't actually an eigenvalue, as only singular matrices have non-trivial solutions.

Q5: How precise are the results?
A: Results are limited by floating-point precision. For exact values, use symbolic computation.