1. What is an Eigenvalue Calculator?

Definition: This calculator solves for the eigenvalues (λ) of a 2x2 matrix by finding the roots of the characteristic polynomial.

Purpose: Eigenvalues are fundamental in linear algebra, used in stability analysis, vibration analysis, and many other applications.

2. How Does the Calculator Work?

The calculator uses the characteristic equation:

Where:

• \( A \) — Input matrix
• \( \lambda \) — Eigenvalue (scalar)
• \( I \) — Identity matrix
• \( \det \) — Determinant of matrix

Explanation: For a 2x2 matrix, this becomes solving the quadratic equation λ² - (trace)λ + (determinant) = 0.

3. Importance of Eigenvalues

Details: Eigenvalues reveal fundamental properties of matrices, including stability of systems, principal components in data, and natural frequencies.

4. Using the Calculator

Tips: Enter all four elements of your 2x2 matrix. The calculator will return real or complex eigenvalues as appropriate.

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex eigenvalues?
A: Complex eigenvalues (with 'i') indicate oscillatory behavior in systems. They're valid results for many real-world applications.

Q2: Can this calculator handle larger matrices?
A: This version is designed for 2x2 matrices. Larger matrices require more sophisticated methods.

Q3: What does a zero eigenvalue mean?
A: A zero eigenvalue indicates the matrix is singular (non-invertible) and has a nontrivial null space.

Q4: How are eigenvalues used in real applications?
A: They're used in principal component analysis (PCA), mechanical vibrations, quantum mechanics, and stability analysis of systems.

Q5: What's the relationship between trace/determinant and eigenvalues?
A: For 2x2 matrices, trace = λ₁ + λ₂ and determinant = λ₁ × λ₂.