Eigen Values Calculator
An eigenvalue is the factor by which a matrix stretches one of its special directions: for a square matrix A, a nonzero vector v with Av = λv keeps its direction and is only scaled by λ. This calculator finds those values for a 2×2 or 3×3 matrix and shows the characteristic polynomial they come from, including complex pairs when the eigenvalues are not real. If you also need the vectors themselves and the diagonalization verdict, use the eigenbasis calculator linked further down; this page is deliberately focused on the eigenvalues alone.
Quick Answer
For the pre-filled matrix [[2, 1], [1, 2]], the eigenvalues are λ1 = 3.0000 and λ2 = 1.0000, from the characteristic polynomial λ² – 4.00λ + 3.00 = 0. Switch to 3×3 and the pre-filled example [[-2, -4, 2], [-2, 1, 2], [4, 2, 5]] gives -5, 3 and 6. Results stay hidden until you click Calculate.
Key takeaways
- Eigenvalues are the roots of the characteristic polynomial det(A – λI) = 0. This tool always shows that polynomial next to the answers, so the result can be checked, not just trusted.
- Two free checks work on every result: the eigenvalues add up to the trace and multiply to the determinant.
- Complex eigenvalues are reported as a conjugate pair (for example 0.0000 + 1.0000i), never as an error or a silent blank.
- This page returns eigenvalues only. Eigenvectors, and the question of whether they form a basis, live on the eigenbasis page.
- Quick Answer
- How to use the eigen values calculator
- Where eigenvalues come from: the characteristic polynomial
- Worked example: eigenvalues of [[2, 1], [1, 2]]
- Worked example: a 3×3 matrix with eigenvalues -5, 3 and 6
- Repeated, complex and zero eigenvalues: reading the badge
- Eigenvalues are half the answer: when you need the eigenbasis instead
- What competing calculators miss
- Where eigenvalues get used
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the eigen values calculator
Pick the matrix size, type the entries, then click Calculate. The grid is pre-filled with a worked example for each size, so the first click shows a complete result you can compare with the worked examples below.
- Matrix size: choose 2×2 or 3×3. Changing size rebuilds the grid with that size’s worked example and hides any old result.
- Entries: type one number per cell, row by row. Decimals and negative numbers are fine; an empty cell counts as 0.
- Calculate: the eigenvalues appear as λ1, λ2 (and λ3 for 3×3), with the characteristic polynomial underneath and a badge naming the result type: real and distinct, repeated, or complex pair.
- Reset: restores the pre-filled worked example for the current size and hides the results.
Where eigenvalues come from: the characteristic polynomial
Eigenvalues are the values of λ that solve det(A – λI) = 0, the characteristic equation described in Wolfram MathWorld’s characteristic polynomial entry. Subtracting λI shifts the diagonal, and the determinant of the shifted matrix is a polynomial in λ whose roots are exactly the eigenvalues. A 2×2 matrix gives a quadratic, a 3×3 gives a cubic, and an n by n matrix gives a degree-n polynomial, which is why an n by n matrix always has n eigenvalues when repeats and complex values are counted.
3×3: λ³ – tr·λ² + M·λ – det = 0, where M is the sum of the three principal 2×2 minors
Checks: λ1 + λ2 (+ λ3) = trace, and λ1 × λ2 (× λ3) = determinant
For 2×2 matrices this tool uses the closed-form quadratic formula, so the answer is exact up to the 4-decimal display rounding. For 3×3 matrices there is no tidy formula worth printing, so the tool isolates each real root of the cubic between its critical points and refines it by bisection until the bracket is under 1e-12 wide: the same bracketing approach numerical libraries document for root isolation, and the reason the NumPy eig documentation also treats eigenvalues as a numerical result rather than a symbolic one. When a 3×3 matrix has only one real eigenvalue, the tool divides the cubic by that root and solves the remaining quadratic, which is where the complex pair comes from. Displayed values are rounded to 4 decimal places; the underlying roots are accurate to far more digits than are shown.
Worked example: eigenvalues of [[2, 1], [1, 2]]
This symmetric matrix is the standard first example for a reason: every step can be checked by hand, and it is the matrix pre-filled in the calculator.
The polynomial factors as (λ – 3)(λ – 1) = 0, which agrees with the quadratic formula. Both checks pass: 3 + 1 = 4, the trace, and 3 × 1 = 3, the determinant. Geometrically, this matrix stretches one diagonal direction by a factor of 3 and the perpendicular diagonal direction by a factor of 1, leaving that second direction completely unchanged.
Worked example: a 3×3 matrix with eigenvalues -5, 3 and 6
Switch the calculator to 3×3 and it loads A = [[-2, -4, 2], [-2, 1, 2], [4, 2, 5]]. The trace is -2 + 1 + 5 = 4. The three principal minors are (-2 × 1) – (-4 × -2) = -10, (-2 × 5) – (2 × 4) = -18, and (1 × 5) – (2 × 2) = 1, which sum to M = -27. The determinant is -90. That gives the cubic the tool displays:
| Check | From the eigenvalues | From the matrix |
|---|---|---|
| Sum of eigenvalues | -5 + 3 + 6 = 4 | Trace = -2 + 1 + 5 = 4 |
| Sum of pairwise products | (-5 × 3) + (-5 × 6) + (3 × 6) = -27 | M = -10 – 18 + 1 = -27 |
| Product of eigenvalues | -5 × 3 × 6 = -90 | Determinant = -90 |
All three checks match, which is what a correct cubic solution looks like. Each root also satisfies the polynomial directly: 6³ – 4(6²) – 27(6) + 90 = 216 – 144 – 162 + 90 = 0.
Repeated, complex and zero eigenvalues: reading the badge
The result badge names which of four situations your matrix falls into. Each one means something different, and the honest handling of the awkward cases is where eigenvalue tools most often cut corners.
| Case | Example matrix | Tool output | What it means |
|---|---|---|---|
| Real and distinct | [[2, 1], [1, 2]] | 3.0000, 1.0000 | Two different stretch factors, two independent directions. |
| Repeated | [[2, 1], [0, 2]] | 2.0000, 2.0000 | One value with algebraic multiplicity 2. Whether it has two independent eigenvectors is a separate question. |
| Complex pair | [[0, -1], [1, 0]] | 0.0000 + 1.0000i, 0.0000 – 1.0000i | The matrix rotates as well as scales, so no real direction survives unchanged. |
| Zero eigenvalue | [[1, 2], [2, 4]] | 5.0000, 0.0000 | The matrix is singular: it flattens one direction to nothing, and its determinant is 0. |
Complex results deserve one extra note. For a real matrix, complex eigenvalues always arrive as a conjugate pair, a + bi and a – bi, because the characteristic polynomial has real coefficients. A rotation matrix is the classic case, and this site has a dedicated complex eigenvalue calculator that works through the algebra of those pairs in more depth. If you need the matching directions for the real cases, each eigenvalue’s direction comes from solving (A – λI)v = 0, and the set of all such directions for one λ is its eigenspace: the basis for eigenspace calculator finds a basis for it, while the complex eigenvector calculator covers the eigenvectors belonging to complex pairs.
Eigenvalues are half the answer: when you need the eigenbasis instead
An eigenvalue tells you the stretch factor but not the direction it acts along. Many questions stop there: stability of a system (do all eigenvalues of the update matrix sit inside the unit circle?), vibration frequencies, whether a matrix is singular, and the variance split in principal component analysis, where the eigenvalues of the covariance matrix rank the components. The eigenvalue treatment in Gilbert Strang’s MIT OpenCourseWare linear algebra course makes the same split: eigenvalues first, then the vectors that complete the picture.
When the question is about directions, powers of a matrix, or diagonalization, you need the vectors as well. The eigenbasis calculator takes the same 2×2 and 3×3 matrices and adds unit eigenvectors for each eigenvalue, states whether they form a basis (the A = PDP⁻¹ verdict, with D named), and flags the two cases where no real eigenbasis exists: a defective matrix with a repeated eigenvalue but too few independent eigenvectors, and a complex pair. Use this page for the values and the polynomial; use that page for the full decomposition story.
What competing calculators miss
Most free eigenvalue tools answer the same question, so the differences are in what surrounds the answer.
- The polynomial disappears: several well-known tools show eigenvalues with no characteristic polynomial, or hide it behind a steps paywall. Without the polynomial you cannot run the trace and determinant checks. Here it is printed on every result, for both sizes.
- Complex pairs get an error message: a common pattern is a warning that the matrix “has complex eigenvalues” instead of the pair itself. This tool computes and displays the pair in both sizes, including the 3×3 case where one real root sits next to a complex pair.
- 3×3 accuracy goes unexplained: tools that solve the cubic rarely say how. Here the method is stated plainly: critical-point bracketing plus bisection to under 1e-12, with the deflated quadratic supplying any complex pair.
- Eigenvectors get bundled in silently: combined eigenvalue-and-eigenvector tools blur which output answers which question, and some charge for the steps. This page does one job, eigenvalues plus polynomial, and points to a separate free page for vectors and diagonalization.
Where eigenvalues get used
Three uses account for most searches for this tool. In principal component analysis, the eigenvalues of a covariance matrix are the variances along the principal components, so their sizes decide how many components carry the signal. In vibration and structural work, the eigenvalues of a stiffness-mass system are the squares of the natural frequencies, and a missed or wrong root means a missed resonance. In stability analysis for discrete systems x(k+1) = Ax(k), the system settles only when every eigenvalue has absolute value under 1, which is why the sign and size of each root matter more than the vectors there. The MathWorks eig documentation covers the same ground from the software side, including the balancing step large-scale solvers apply before computing.
Frequently asked questions
An eigenvalue is a scalar λ for which a nonzero vector v satisfies Av = λv: the matrix leaves v pointing in its original direction and only stretches it by the factor λ. A negative eigenvalue flips the direction, an eigenvalue of 1 leaves the vector unchanged, and an eigenvalue of 0 means the matrix flattens some direction completely, so the matrix is singular.
It uses the trace and determinant in the exact quadratic formula. The characteristic polynomial of a 2×2 matrix is λ² – tr·λ + det = 0, so the eigenvalues are (tr + √(tr² – 4det)) / 2 and (tr – √(tr² – 4det)) / 2. For [[2, 1], [1, 2]] the trace is 4 and the determinant is 3, giving λ = 3 and λ = 1 with no numerical approximation involved.
The tool builds the cubic λ³ – tr·λ² + M·λ – det = 0, where M is the sum of the principal 2×2 minors, then isolates each real root between the cubic’s critical points and refines it by bisection until the bracket is under 1e-12 wide. If only one real root exists, the cubic is divided by that root and the remaining quadratic gives the complex pair. Results are displayed to 4 decimal places, and the trace, minor and determinant checks printed in the worked example confirm the roots.
Yes. A real matrix can have complex eigenvalues, and they always arrive as a conjugate pair a + bi and a – bi because the characteristic polynomial has real coefficients. The rotation matrix [[0, -1], [1, 0]] gives 0.0000 + 1.0000i and 0.0000 – 1.0000i in this calculator. Complex eigenvalues signal rotation in the transformation, and such a matrix has no real eigenvectors.
A repeated eigenvalue is a root that appears more than once in the characteristic polynomial; its count is its algebraic multiplicity. The matrix [[2, 1], [0, 2]] has λ = 2 twice. A repeat on its own is not a problem, but it raises a second question the eigenvalues cannot answer: whether the matrix still has enough independent eigenvectors. That is the defective-matrix case covered by the eigenbasis calculator.
The characteristic polynomial is det(A – λI) written out as a polynomial in λ: a quadratic for a 2×2 matrix and a cubic for a 3×3. Its roots are the eigenvalues. The calculator shows it on every result so you can verify the answer yourself: the eigenvalues must add up to the trace and multiply to the determinant, and each one must make the polynomial equal zero.
No. This page returns eigenvalues and the characteristic polynomial only, by design. For unit eigenvectors for each eigenvalue, the diagonal matrix D, and a verdict on whether the eigenvectors form an eigenbasis (A = PDP⁻¹), use the eigenbasis calculator on this site, which takes the same 2×2 and 3×3 matrices and also reports defective and complex cases honestly.
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References
- Wolfram MathWorld. Eigenvalue. Accessed October 21, 2026.
- Wolfram MathWorld. Characteristic Polynomial. Accessed October 21, 2026.
- Wolfram MathWorld. Eigenvector. Accessed October 21, 2026.
- Strang, G., MIT OpenCourseWare. Linear Algebra (18.06SC). Accessed October 21, 2026.
- MathWorks. eig: Eigenvalues and Eigenvectors. Accessed October 21, 2026.
- NumPy Developers. numpy.linalg.eig. Accessed October 21, 2026.