Linear Algebra
Eigenvalue Calculator
Inspect the eigenvalues, eigenvectors, trace, and determinant of a 2×2 matrix in one panel.
Compute eigenvalues and normalized eigenvectors for a 2×2 matrix.
Characteristic equation
λ² − (a + d)λ + (ad − bc) = 0
Eigenvalues solve the characteristic polynomial. For each eigenvalue, the calculator returns a normalized eigenvector.
How to use
- Enter a 2×2 matrix with two numbers per row.
- Submit to compute eigenvalues λ₁ and λ₂ along with representative eigenvectors.
- Review supporting information like trace and determinant.
Example
Input: Matrix = [[4,1],[2,3]]
Output: Eigenvalues ≈ 5.5616 and 1.4384 with normalized eigenvectors.
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Eigenvalue Calculator: matrix calculator, determinant calculator, matrix multiplication calculator, eigenvalue calculator. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Eigenvalue Calculator: Finds eigenvalues and normalized eigenvectors for 2×2 matrices. It's built around eigenvalue, eigenvector, characteristic polynomial, so you can go from a raw question to a checked answer without switching tools.
The math behind it: Eigenvalues solve the characteristic polynomial. For each eigenvalue, the calculator returns a normalized eigenvector. The core relationship is λ² − (a + d)λ + (ad − bc) = 0, shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Enter a 2×2 matrix with two numbers per row. (2) Submit to compute eigenvalues λ₁ and λ₂ along with representative eigenvectors. (3) Review supporting information like trace and determinant. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering Matrix = [[4,1],[2,3]] returns Eigenvalues ≈ 5.5616 and 1.4384 with normalized eigenvectors.. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
Does it handle complex eigenvalues?
When the discriminant is negative the calculator reports the repeated real component. Complex visualization is planned for a future update.
Can I analyse larger matrices?
This tool focuses on 2×2 matrices for clarity. Use the polynomial solver for higher-degree characteristic equations.
What formula does the Eigenvalue Calculator use?
Eigenvalues solve the characteristic polynomial. For each eigenvalue, the calculator returns a normalized eigenvector.
How do I use the Eigenvalue Calculator?
Enter a 2×2 matrix with two numbers per row. Submit to compute eigenvalues λ₁ and λ₂ along with representative eigenvectors. Review supporting information like trace and determinant.