Matrix Input
Example Data Table
| Matrix | Trace | Determinant | Expected Eigenvalues |
|---|---|---|---|
| [[4, 2], [1, 3]] | 7 | 10 | 5, 2 |
| [[2, 0], [0, 6]] | 8 | 12 | 2, 6 |
| [[4, 2, 0], [1, 3, 0], [0, 0, 5]] | 12 | 50 | 5, 5, 2 |
Formula Used
Eigenvalues are values of λ that make the matrix equation A v = λ v true for a nonzero vector v.
The main equation is det(A - λI) = 0. This is called the characteristic equation.
For a 2 x 2 matrix, the equation can be written as λ² - trace(A)λ + det(A) = 0.
For a 3 x 3 matrix, the equation can be written as λ³ - trace(A)λ² + Sλ - det(A) = 0.
Here, S is the sum of the three principal 2 x 2 minors.
After each eigenvalue is found, an eigenvector candidate is calculated from (A - λI)v = 0.
How to Use This Calculator
Select a 2 x 2 or 3 x 3 matrix.
Enter every matrix cell in the input boxes.
Use decimals, whole numbers, or negative values.
Press the calculate button to show results above the form.
Review the trace, determinant, polynomial, eigenvalues, and vector candidates.
Use the CSV or PDF button to save the calculation summary.
Understanding Matrix Eigenvalues
Eigenvalues describe how a square matrix stretches or reverses special directions. Those directions are eigenvectors. When a matrix multiplies an eigenvector, the output stays on the same line. Only the length and sign change. The scale factor is the eigenvalue. This idea is central in algebra, vibration study, stability testing, graphics, statistics, and differential equations.
Why This Calculator Helps
Manual eigenvalue work can become long very quickly. A two by two matrix uses a quadratic equation. A three by three matrix uses a cubic equation. This calculator keeps the steps visible. It reports trace, determinant, the characteristic polynomial, eigenvalues, and practical eigenvector candidates. These values help students check homework and help professionals review models before deeper software analysis.
Interpreting the Output
Real eigenvalues show direct stretching along real directions. Negative values show reversal with scaling. Complex eigenvalues often indicate rotation combined with scaling. Repeated eigenvalues need careful review because they may not provide enough independent eigenvectors. The determinant equals the product of eigenvalues. The trace equals their sum when multiplicity is counted. These checks are useful for spotting input mistakes.
Good Input Practice
Use exact entries when possible. Decimals are accepted, but rounded decimals can slightly change roots. Enter every cell of the square matrix. Select the correct matrix size before calculating. If a result is very close to zero, treat it as a numerical zero unless your problem requires strict precision. Export the summary when you need to attach steps to a report, worksheet, or project record.
Where Eigenvalues Appear
In engineering, eigenvalues describe natural frequencies and buckling behavior. In data science, they support principal component analysis and covariance review. In Markov chains, they explain long term transition behavior. In finance, they can expose hidden structure in correlation matrices. In computer graphics, they help analyze transformations and repeated scaling. In education, they connect determinants, systems, polynomials, and vectors in one topic. This calculator is designed for quick exploration, not symbolic proof. Always compare important results with course requirements or trusted software when exact algebraic forms are required. The displayed eigenvectors are convenient candidates, so any nonzero scalar multiple represents the same direction. Small rounding differences are normal when roots involve complex or repeated behavior in reports.
FAQs
What is an eigenvalue?
An eigenvalue is a scale factor. It shows how a matrix changes a special nonzero vector without changing that vector direction.
What matrix sizes are supported?
This calculator supports 2 x 2 and 3 x 3 square matrices. These are common sizes for classroom and quick applied calculations.
Can eigenvalues be complex?
Yes. Some real matrices have complex eigenvalues. Complex values often appear when a matrix represents rotation with scaling.
What is the characteristic polynomial?
It is the polynomial made from det(A - λI). Its roots are the eigenvalues of the matrix.
Why is the determinant shown?
The determinant is useful because it equals the product of eigenvalues when multiplicity is included.
Why is the trace shown?
The trace is useful because it equals the sum of eigenvalues when multiplicity is included.
Are eigenvector candidates unique?
No. Any nonzero scalar multiple of an eigenvector points in the same eigen direction and is also valid.
Can I use decimals?
Yes. Decimal entries are accepted. Small rounding differences can appear, especially with repeated or complex roots.