Enter A and B matrices for a precise solver. Review determinants, traces, residuals, and conditioning. Download clean outputs for homework, labs, simulations, and audits.
This solver handles real 2×2 and 3×3 generalized eigenvalue systems of the form A x = λ B x.
| Case | Matrix A | Matrix B | Expected finite λ values | Physics note |
|---|---|---|---|---|
| Coupled oscillator | [[8, -2], [-2, 6]] | [[2, 0], [0, 1]] | 3, 5 | For λ = ω², angular frequencies are √3 and √5. |
| Three degree model | [[8, -2, 0], [-2, 6, 0], [0, 0, 4]] | [[2, 0, 0], [0, 1, 0], [0, 0, 1]] | 3, 4, 5 | Useful for stiffness and mass matrix studies. |
The generalized eigenvalue problem is written as A x = λ B x.
Finite eigenvalues are found from the characteristic equation det(A - λB) = 0.
After each eigenvalue is computed, the solver builds the matrix A - λB.
It then finds a nonzero vector in that matrix null space.
This vector is the corresponding eigenvector.
The solver normalizes each eigenvector to unit length.
It also evaluates the residual norm ||(A - λB)x||.
A smaller residual means a more consistent numerical result.
When λ is real and nonnegative, many physics models use λ = ω².
In that case, the angular frequency is ω = √λ.
The cyclic frequency is f = ω / (2π).
Choose either a 2×2 or 3×3 system size.
Enter the entries of matrix A.
Enter the entries of matrix B.
Set the decimal precision you want in the output.
Click Solve System.
The result appears above the form and below the header section.
Review the characteristic polynomial, finite eigenvalues, normalized eigenvectors, and residual norms.
If your model represents vibration or wave motion, use the ω and Hz columns for direct physical interpretation.
Use the CSV button for spreadsheet analysis.
Use the PDF button for a print friendly export.
Generalized eigenvalue problems appear in vibration theory, quantum models, elastic systems, and wave equations. Many physical models use a stiffness matrix and a mass matrix. The eigenvalues reveal natural modes. The eigenvectors describe the associated shapes. This solver is designed for those small but important matrix problems.
Standard eigenvalue solvers use one matrix. A generalized solver uses two. That difference matters in physics. The second matrix often stores mass, overlap, inertia, or metric information. Because of that, the equation A x = λ B x gives more realistic mode behavior for coupled systems.
The tool computes the finite roots of det(A − λB) = 0. It also reports normalized eigenvectors. Residual norms help you judge numerical quality. The determinant values provide a fast consistency check. When B is invertible, the solver also reports trace(B-1A) and det(B-1A). These values are useful in analytical verification.
In many mechanics and field problems, λ equals ω². That means positive real eigenvalues produce usable angular frequencies. The calculator converts those values into ω and frequency in hertz. This saves time during lab work, homework checks, and model validation.
Use this page when you need a quick, transparent solver for 2×2 or 3×3 systems. It is ideal for teaching, prototyping, and sanity checks before larger numerical runs. The plain layout also makes the output easy to export, compare, and document.
It solves real 2×2 or 3×3 generalized eigenvalue systems written as A x = λ B x. It returns finite eigenvalues, normalized eigenvectors, and residual norms.
Matrix B often represents mass, overlap, or a metric. It changes the physical meaning of the modes and makes the model more realistic than a standard one-matrix problem.
Yes. Many vibration models use λ = ω². For nonnegative real eigenvalues, the calculator also reports angular frequency and hertz values for direct interpretation.
It measures how well each reported eigenpair satisfies the equation. Smaller residuals indicate a stronger numerical match between the matrices, eigenvalue, and eigenvector.
A singular B can produce infinite eigenvalues. This page lists the finite roots it can compute and warns that additional infinite modes may exist.
Yes. The solver can report complex eigenvalues. Their eigenvectors are also computed numerically and shown in normalized complex form when needed.
Normalization makes mode shapes easier to compare. It also produces cleaner exported results and improves consistency when reviewing different solutions side by side.
Use CSV for analysis in spreadsheet tools. Use PDF when you want a clean printout for notes, reports, homework submissions, or project records.