1. What is an Eigenvector?

An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it. For a matrix A, an eigenvector v satisfies the equation Av = λv, where λ is the eigenvalue.

2. How Does the Calculator Work?

The calculator solves the eigenvector equation:

Where:

• \( A \) — Input matrix
• \( \lambda \) — Eigenvalue
• \( I \) — Identity matrix
• \( \vec{v} \) — Eigenvector to be found

Explanation: The calculator finds a non-trivial solution to the homogeneous system of equations formed by (A - λI)v = 0.

3. Importance of Eigenvectors

Details: Eigenvectors are fundamental in many areas including stability analysis, quantum mechanics, vibration analysis, and principal component analysis (PCA).

4. Using the Calculator

Tips: Enter a 2x2 matrix and one of its eigenvalues. The calculator will return a corresponding normalized eigenvector.

5. Frequently Asked Questions (FAQ)

Q1: Can this calculator handle complex eigenvalues?
A: No, this calculator only works with real eigenvalues and eigenvectors.

Q2: What if my matrix is larger than 2x2?
A: This calculator is designed for 2x2 matrices only. For larger matrices, more sophisticated methods are needed.

Q3: Why is the eigenvector normalized?
A: Normalization (to unit length) provides a standard representation, as eigenvectors are only defined up to a scalar multiple.

Q4: What if I get [0,0] as result?
A: This indicates an error - either your input wasn't a valid eigenvalue, or there was a calculation error.

Q5: Can I find all eigenvectors at once?
A: This calculator finds one eigenvector at a time. For complete eigenanalysis, you would need to input each eigenvalue separately.