1. What is an Eigenvector?

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

2. How Does the Calculator Work?

The calculator solves the equation:

Where:

• \( A \) — Square matrix
• \( \lambda \) — Eigenvalue (scalar)
• \( I \) — Identity matrix of same size as A
• \( v \) — Eigenvector (non-zero solution)

Explanation: The equation finds vectors that are scaled but not rotated when the matrix transformation is applied.

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 the matrix in format "a,b;c,d" for 2x2 matrices. The eigenvalue must be a known eigenvalue of the matrix. The calculator will find the corresponding eigenvector.

5. Frequently Asked Questions (FAQ)

Q1: Can any matrix have eigenvectors?
A: Only square matrices have eigenvectors and eigenvalues. Not all square matrices have a full set of eigenvectors.

Q2: What's the difference between left and right eigenvectors?
A: Right eigenvectors satisfy Av = λv, while left eigenvectors satisfy vᵀA = λvᵀ. For symmetric matrices, they are the same.

Q3: Can a matrix have multiple eigenvectors for one eigenvalue?
A: Yes, the set of eigenvectors for a particular eigenvalue forms a subspace called the eigenspace.

Q4: Are eigenvectors always real numbers?
A: No, matrices with complex eigenvalues will have complex eigenvectors.

Q5: How are eigenvectors used in real-world applications?
A: They're used in Google's PageRank algorithm, facial recognition, mechanical vibration analysis, and many other areas.