1. What is the Eigenvector Equation?

The eigenvector equation \((A - \lambda I)v = 0\) is fundamental in linear algebra for finding the eigenvectors of a matrix given its eigenvalues. An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it.

2. How Does the Calculator Work?

The calculator uses the eigenvector equation:

Where:

• \( A \) — Square matrix
• \( \lambda \) — Eigenvalue of matrix A
• \( I \) — Identity matrix of same size as A
• \( v \) — Eigenvector corresponding to λ

Explanation: The equation represents a homogeneous system of linear equations. The solution space gives the eigenvectors corresponding to the eigenvalue λ.

3. Importance of Eigenvectors

Details: Eigenvectors are crucial in many areas including stability analysis, quantum mechanics, vibration analysis, facial recognition algorithms, and principal component analysis in statistics.

4. Using the Calculator

Tips: Enter the matrix in format "a,b;c,d" for 2x2 matrices. For larger matrices, add more rows separated by semicolons and columns by commas. The eigenvalue must be a known eigenvalue of the matrix.

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix isn't square?
A: Only square matrices have eigenvalues and eigenvectors. The calculator will verify the matrix is square.

Q2: How do I know if my λ is actually an eigenvalue?
A: λ must satisfy det(A - λI) = 0. The calculator assumes you've already verified this.

Q3: What if there are multiple eigenvectors?
A: The eigenspace may have dimension >1. The calculator will return a basis for the eigenspace.

Q4: Can I use complex numbers?
A: This calculator handles real numbers only. For complex eigenvalues/vectors, specialized tools are needed.

Q5: What's the largest matrix size supported?
A: For practical reasons, matrices larger than 5x5 may not be processed efficiently in this implementation.