Enter paired matrices
Use a 2×2 or 3×3 matrix pencil. The form stays single-column overall, while fields flow into responsive columns.
Example data table
This sample 2×2 pencil produces two clean real generalized eigenvalues.
| A11 | A12 | A21 | A22 | B11 | B12 | B21 | B22 | Expected roots |
|---|---|---|---|---|---|---|---|---|
| 6 | 2 | 1 | 3 | 2 | 0 | 0 | 1 | 2, 4 |
Formula used
The calculator solves the generalized eigenvalue problem Ax = λBx by building the matrix pencil A - λB.
The finite generalized eigenvalues are the roots of det(A - λB) = 0. For 2×2 and 3×3 inputs, the page builds that polynomial, trims near-zero leading terms, and solves the resulting linear, quadratic, or cubic equation.
If B is invertible, the same values are the ordinary eigenvalues of B-1A. When B is singular, some eigenvalues may be infinite, which is why the page also reports a reduced polynomial degree warning.
How to use this calculator
- Select either a 2×2 or 3×3 matrix dimension.
- Enter the coefficients of matrix A and matrix B.
- Choose the display precision for the reported numbers.
- Press the calculate button to place the results above the form.
- Review the polynomial, determinants, roots, and interpretation notes.
- Use the CSV or PDF buttons to save the current analysis.
Frequently asked questions
1) What is a generalized eigenvalue?
It is a scalar λ that satisfies Ax = λBx for some nonzero vector x. This extends the ordinary eigenvalue problem by introducing a second matrix.
2) Why does the calculator ask for two matrices?
Generalized eigenvalues come from a matrix pair, often called a matrix pencil. Matrix A supplies the main system, while matrix B changes the scaling or constraint structure.
3) What does det(A - λB) = 0 mean?
It is the characteristic equation for the matrix pair. When the determinant becomes zero, the pencil loses rank, allowing a nonzero vector solution for the generalized eigenvalue relation.
4) Why can some roots be complex?
Real matrices can still produce complex conjugate roots. This happens when the characteristic polynomial has a negative discriminant or an irreducible quadratic factor.
5) What if matrix B is singular?
A singular B can create infinite generalized eigenvalues or lower the polynomial degree. The calculator flags this situation so you can interpret the finite roots more carefully.
6) Does this calculator support large matrices?
This page is intentionally focused on 2×2 and 3×3 inputs. That keeps the algebra transparent, the polynomial explicit, and the reported diagnostics easy to verify manually.
7) When is B-1A useful?
If B is invertible, generalized eigenvalues equal the ordinary eigenvalues of B-1A. That view is convenient for theory, but the determinant method still works directly.
8) What do the export buttons save?
They save the current inputs and computed summary values. The CSV export is good for spreadsheets, while the PDF export creates a shareable report snapshot.