Calculator Use
The eigenvalue and eigenvector calculator is a specialized tool designed to solve the characteristic equation of a 2×2 square matrix. This tool is essential for students and engineers working with linear transformations, structural analysis, and vibration modeling. By entering the four components of a matrix, you can instantly find the scalars (eigenvalues) and the associated directions (eigenvectors) that remain unchanged in direction during a linear transformation.
- a11, a12, a21, a22
- These represent the elements of the square matrix A. In a standard 2×2 grid, a11 is top-left, a12 is top-right, a21 is bottom-left, and a22 is bottom-right.
- Show Steps
- Check this box to view the derived characteristic quadratic equation before the final results.
How It Works
To find the eigenvalues, the calculator solves the equation det(A – λI) = 0, where I is the identity matrix and λ represents the eigenvalues. For a 2×2 matrix, this results in a quadratic equation known as the characteristic equation:
λ² – tr(A)λ + det(A) = 0
- tr(A) (Trace): The sum of the diagonal elements (a11 + a22).
- det(A) (Determinant): The value (a11 * a22) – (a12 * a21).
- Eigenvectors: Once λ is found, the calculator solves (A – λI)v = 0 to find the non-zero vector v.
Calculation Example
Example: Find the eigenvalues and eigenvectors for the matrix A = [[4, 1], [2, 3]].
Step-by-step solution:
- Identify Values: a11=4, a12=1, a21=2, a22=3.
- Calculate Trace: 4 + 3 = 7.
- Calculate Determinant: (4 * 3) – (1 * 2) = 12 – 2 = 10.
- Characteristic Equation: λ² – 7λ + 10 = 0.
- Solve for λ: (λ – 5)(λ – 2) = 0. So, λ₁ = 5, λ₂ = 2.
- Result: The eigenvalues are 5 and 2. The calculator then computes the normalized eigenvectors based on these values.
Common Questions
What is an eigenvalue?
An eigenvalue is a scalar that indicates how much a corresponding eigenvector is stretched or compressed during a linear transformation. If the eigenvalue is negative, the direction of the vector is reversed.
Why are eigenvectors normalized?
Eigenvectors represent a direction. Since any scalar multiple of an eigenvector is also an eigenvector, this calculator provides a "normalized" version (a unit vector with a length of 1) for consistency and ease of use in further physics or engineering calculations.
Can a matrix have complex eigenvalues?
Yes. If the discriminant of the characteristic equation is negative, the eigenvalues will be complex numbers. This usually indicates a rotation in the transformation rather than just a simple scaling.