1. What are Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that only gets scaled (not rotated) when A is applied to it. The scaling factor is the eigenvalue λ.

2. How the Calculator Works

The calculator solves the characteristic equation:

Where:

• \( A \) — Input 4×4 matrix
• \( \lambda \) — Eigenvalues (roots of the characteristic polynomial)
• \( I \) — Identity matrix of same size as A
• \( \det \) — Matrix determinant

For each eigenvalue λ, the corresponding eigenvector v is found by solving: \[ (A - \lambda I)\mathbf{v} = 0 \]

3. Mathematical Background

Details: For a 4×4 matrix, the characteristic equation is a quartic polynomial. The calculator uses numerical methods to find all roots (real and complex) and their corresponding eigenvectors.

4. Using the Calculator

Tips: Enter all 16 elements of your 4×4 matrix. The calculator will display eigenvalues and their corresponding eigenvectors. Results may be real or complex numbers.

5. Frequently Asked Questions (FAQ)

Q1: What if my matrix has complex eigenvalues?
A: The calculator will display both real and complex eigenvalues and eigenvectors if they exist.

Q2: Can a matrix have repeated eigenvalues?
A: Yes, some matrices have eigenvalues with multiplicity greater than 1 (called algebraic multiplicity).

Q3: What's the difference between left and right eigenvectors?
A: This calculator finds right eigenvectors (Av = λv). Left eigenvectors satisfy vᵀA = λvᵀ.

Q4: Why are eigenvectors important?
A: Eigenvectors are crucial in many applications including stability analysis, principal component analysis (PCA), and quantum mechanics.

Q5: What if my matrix is not diagonalizable?
A: Non-diagonalizable matrices (defective matrices) don't have a complete set of eigenvectors. In such cases, generalized eigenvectors can be used.