Find Eigenvalues of Matrix Calculator

Calculator Inputs

Example Data Table

Formula Used

Eigenvalues are found from the characteristic equation:

det(λI - A) = 0

Here, A is the square matrix, I is the identity matrix, and λ is an eigenvalue. The code forms this polynomial with the Faddeev-LeVerrier method. It then solves the polynomial with the Durand-Kerner root method. The trace equals the sum of eigenvalues. The determinant equals their product.

How to Use This Calculator

1. Select the matrix order from 2 x 2 to 6 x 6.
2. Enter every visible matrix value.
3. Choose decimal places, tolerance, iterations, and sorting.
4. Press the calculate button.
5. Read the eigenvalue table above the form.
6. Use the CSV or PDF button to save results.

Understanding Eigenvalues

Eigenvalues reveal how a square matrix stretches a vector without changing its direction. They appear in linear algebra, differential equations, vibration analysis, graphics, statistics, and machine learning. This calculator focuses on real square matrices and returns complex values when the characteristic equation needs them.

Why This Calculator Helps

Manual eigenvalue work becomes slow when matrix order increases. A small sign error can change every root. This tool builds the characteristic polynomial first. Then it solves the polynomial numerically and checks each root. You can review the trace, determinant, polynomial, root list, and residual error in one place.

Advanced Matrix Support

You can choose matrix sizes from two to six. Enter decimals, negative values, or zero values. The calculator accepts general square matrices, not only diagonal or symmetric matrices. It also lets you control precision, tolerance, maximum iterations, and sorting. These options help when values are close together or when a matrix is poorly scaled.

Interpreting Results

A real eigenvalue has no imaginary part. A complex eigenvalue is shown with i. For real matrices, complex values usually appear in conjugate pairs. The trace check compares the sum of eigenvalues with the matrix trace. The determinant check compares the product of eigenvalues with the matrix determinant. Small differences may appear because numerical methods use approximation.

Practical Uses

Engineers use eigenvalues to study stability, resonance, and systems. Data analysts use them in principal component analysis. Students use them to verify homework and understand characteristic equations. Programmers use them in transformations and simulations.

Good Input Habits

Start with small matrices while learning. Check that every row has the same number of entries. Avoid rounding too early. Use more decimals when entries are close or very large. If results seem unstable, increase iterations or reduce tolerance. Compare the residual column. A smaller residual means the eigenvalue satisfies the polynomial more closely.

Final Notes

Eigenvalues describe core behavior inside a matrix. They turn a large matrix problem into meaningful scalar values. With clear outputs and downloads, this page supports study, reports, and repeated classroom examples. The calculator is also useful when comparing textbook answers with software output. It keeps the polynomial visible, so each numeric root stays connected to the original matrix more clearly.

FAQs