Eigenvalue Calculator (2×2 Matrix)
Enter the four entries of a 2×2 matrix.
Your numbers
Your results
Eigenvalues
Eigenvalues
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Eigenvector for the first eigenvalue
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Eigenvector for the second eigenvalue
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Trace (sum of eigenvalues)
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Determinant (product of eigenvalues)
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Eigenvalue Calculator (2×2 Matrix) by TheCalcTools
How the Eigenvalue Calculator (2×2 Matrix) works
Eigenvalues solve the characteristic equation det(A − λI) = 0, which for a 2×2 matrix is λ² − (trace)λ + det = 0, so λ = tr/2 ± √(tr²/4 − det).
For each eigenvalue, an eigenvector v satisfies (A − λI)v = 0; for a 2×2 matrix (b, λ − a) works when b ≠ 0. Any non-zero multiple is also an eigenvector.
The formula
λ = tr/2 ± √(tr²/4 − det)
Worked example
Using the default values shown in the calculator:
| a (row 1, column 1) | 4 |
|---|---|
| b (row 1, column 2) | 1 |
| c (row 2, column 1) | 2 |
| d (row 2, column 2) | 3 |
- For [[4, 1], [2, 3]]: λ = 5 and 2.
- Eigenvectors (1, 1) and (1, -2).
Result:
| Eigenvalues | λ = 5 and 2 |
|---|---|
| Eigenvector for the first eigenvalue | (1, 1) |
| Eigenvector for the second eigenvalue | (1, -2) |
| Trace (sum of eigenvalues) | 7.0000 |
| Determinant (product of eigenvalues) | 10.0000 |
Frequently asked questions
What do complex eigenvalues mean?
The matrix rotates vectors: no real direction is simply stretched.
Sources and further reading
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This calculator gives estimates for information only. Check important results independently before you rely on them. See our disclaimer.