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 changes by a scalar factor λ when A is applied to it: Av = λv. The scalar λ is called the eigenvalue.

2. How Does the Calculator Work?

The calculator uses the characteristic equation for 2x2 matrices:

Where:

• \( \text{tr}(A) \) — Trace of matrix A (sum of diagonal elements)
• \( \text{det}(A) \) — Determinant of matrix A
• \( \lambda \) — Eigenvalues of matrix A

Explanation: The equation is derived from solving the characteristic polynomial det(A - λI) = 0.

3. Importance of Eigenvalues

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

4. Using the Calculator

Tips: Enter all four elements of your 2x2 matrix. The calculator will compute the trace, determinant, eigenvalues, and corresponding eigenvectors.

5. Frequently Asked Questions (FAQ)

Q1: What if I get complex eigenvalues?
A: Complex eigenvalues occur when the discriminant is negative. They come in complex conjugate pairs and indicate rotational components in the transformation.

Q2: Can a matrix have only one eigenvalue?
A: Yes, if the matrix has repeated eigenvalues (discriminant = 0), though there may still be two independent eigenvectors.

Q3: What does a zero eigenvalue mean?
A: A zero eigenvalue means the matrix is singular (determinant = 0) and has a non-trivial null space.

Q4: How are eigenvectors useful?
A: Eigenvectors indicate directions that remain unchanged under the linear transformation represented by the matrix.

Q5: Can all matrices be diagonalized?
A: Only if there are enough linearly independent eigenvectors (n for an n×n matrix). Defective matrices cannot be diagonalized.