Complex Eigenvector Calculator
Most eigenvalue tools stop at the eigenvalues and leave the vectors, the one part you usually need, as an exercise. This calculator takes any real 2×2 matrix and returns both eigenvalues and their eigenvectors, including the complex case: when the discriminant is negative, the eigenvalues come out as a conjugate pair a ± bi with matching complex eigenvectors.
It also shows the trace, determinant, discriminant and characteristic polynomial behind the answer, then proves the result by checking A·v = λv for each pair and reporting the residual.
Eigenvectors for real eigenvalues are shown normalized to unit length. Any nonzero scalar multiple of an eigenvector is also a valid eigenvector for the same eigenvalue, so a textbook answer that is a multiple of the one shown here is equally correct.
Quick Answer
Key takeaways
- An eigenvector is a direction the matrix stretches without rotating; the eigenvalue is the stretch factor, and the pair satisfies A·v = λv.
- For a 2x2 matrix everything runs through three numbers: trace, determinant, and the discriminant Δ = trace² − 4·det.
- Δ positive means two distinct real eigenvalues, Δ zero means one repeated eigenvalue, and Δ negative means a complex conjugate pair with no real eigenvectors.
- Eigenvectors are never unique: any nonzero multiple works. This tool normalizes real eigenvectors to unit length so answers are easy to compare.
- Quick Answer
- How to use the complex eigenvector calculator
- The formula behind the calculator
- Reading the discriminant: the three cases
- Worked examples, verified against the tool
- Why complex eigenvalues always come in conjugate pairs
- Where eigenvectors show up in practice
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
How to use the complex eigenvector calculator
Enter the four entries of your matrix in reading order: a₁₁ and a₁₂ across the top row, a₂₁ and a₂₂ across the bottom. Decimals and negative values are accepted, so matrices like [[0, −1], [1, 0]] or [[1.5, 0.25], [−0.5, 2]] work exactly as written. Press Calculate and the results box gives you both eigenvalue and eigenvector pairs first, then the working underneath: trace, determinant, discriminant, the characteristic polynomial, and a plain classification of which of the three cases you are in.
Read the verification line before you trust any answer, here or anywhere else. The calculator substitutes each pair back into A·v = λv using full complex arithmetic and reports the residual. A residual of 0, or a tiny number in scientific notation, means the pair genuinely solves the equation. If you only need the eigenvalues themselves, including for larger problems, the complex eigenvalue calculator covers that half of the problem on its own.
The formula behind the calculator
Eigenvalues are the roots of the characteristic polynomial, and for a 2x2 matrix that polynomial is built from just two quantities: the trace (the sum of the diagonal) and the determinant. That is what makes the 2x2 case solvable in closed form with the quadratic formula, with no iteration and no approximation beyond ordinary rounding.
Two identities give you a free check on any 2x2 answer, at any size of matrix in fact: the eigenvalues always sum to the trace, and they always multiply to the determinant. In the worked example below, 3 + 1 = 4 matches the trace and 3 × 1 = 3 matches the determinant. If a tool, a textbook, or your own hand calculation fails either identity, something is wrong before you even look at the vectors.
Finding the vector is a null space problem: for each eigenvalue, the eigenvectors are exactly the nonzero solutions of (A − λI)v = 0. The full set of those solutions for one eigenvalue is its eigenspace, and if you need a basis for that space written out, the basis for eigenspace calculator and the basis for null space calculator take that step separately.
Reading the discriminant: the three cases
One number decides the shape of the whole answer. The table below is the classification the calculator prints with every result.
| Discriminant | Eigenvalues | What it means geometrically |
|---|---|---|
| Δ > 0 | Two distinct real values | The matrix stretches along two independent real directions Two real eigenvectors |
| Δ = 0 | One repeated real value | Usually still two eigenvectors, as with a multiple of the identity; a few defective matrices have only one Check the vectors |
| Δ < 0 | Complex conjugate pair a ± bi | Rotation is involved, so no real direction survives unchanged No real eigenvectors |
Worked examples, verified against the tool
The calculator is pre-filled with the real case, and every figure below is what it returns when you press Calculate.
| Quantity | Value | How it is obtained |
|---|---|---|
| Trace | 4.0000 | 2 + 2 |
| Determinant | 3.0000 | 2 × 2 − 1 × 1 |
| Discriminant Δ | 4.0000 | 4² − 4 × 3 |
| Characteristic polynomial | λ² − 4.0000λ + 3.0000 = 0 | λ² − Tλ + D |
| λ₁ and v₁ | 3.0000, [0.7071, 0.7071] | (4 + 2) / 2, then normalized |
| λ₂ and v₂ | 1.0000, [0.7071, −0.7071] | (4 − 2) / 2, then normalized |
The complex case is worth running yourself, because it is the case this page exists for. Enter [[0, −1], [1, 0]], the matrix that rotates every vector by 90 degrees. The trace is 0, the determinant is 1, and Δ = −4, so the eigenvalues are λ = ±i, printed by the tool as 1.0000i and −1.0000i. The eigenvectors come out as v₁ = [−1.0000, 1.0000i] and v₂ = [−1.0000, −1.0000i], a conjugate pair, exactly as the theory requires. The verification still holds with complex arithmetic: A·v₁ = [−i, −1], which is i times v₁.
Why complex eigenvalues always come in conjugate pairs
The matrix entries are real, so the characteristic polynomial has real coefficients, and a standard result about real polynomials says their non-real roots always arrive in conjugate pairs: if a + bi is a root, so is a − bi. The eigenvectors inherit the pairing, which is why the rotation example above produces v and its conjugate rather than two unrelated complex vectors.
Geometrically, a complex pair is the algebra telling you the transformation rotates. A real eigenvector would be a direction the matrix leaves pointing the same way, and a pure rotation has no such direction, so no real eigenvector can exist. When a complex pair has a nonzero real part as well, the motion is a spiral: each application rotates and scales by the magnitude of the eigenvalue. That is why complex eigenvalues are the working language of vibration analysis, AC circuit analysis, and stability checks for dynamical systems, where the real part decides whether oscillations grow or decay.
Where eigenvectors show up in practice
Principal component analysis compresses data along the eigenvectors of a covariance matrix, keeping the directions that carry the most variance. Google's PageRank ranks pages by the dominant eigenvector of the web's link matrix. Vibration engineers read a structure's natural frequencies from eigenvalues, and quantum mechanics treats measurable quantities as eigenvalues of operators. In every one of those applications, the pattern from this page is the same: the eigenvector names the natural direction of the system, and the eigenvalue says how strongly the system responds along it. A symmetric matrix, the kind behind covariance and many physical systems, is the friendly case: its eigenvalues are always real and its eigenvectors can be chosen orthogonal, which the discriminant confirms for the pre-filled example.
What competing calculators miss
The top eigenvalue tools reviewed for this page are strong on steps for real matrices, but the complex case and the proof are where they thin out.
Miss 1: complex pairs printed as raw decimals or skipped. Several tools either restrict themselves to real eigenvalues or bury a complex answer in unformatted output. Here the pair is printed cleanly as a ± bi, with conjugate eigenvectors to match, because rotation matrices are a standard homework and engineering case, not an edge case.
Miss 2: no verification. Most tools print numbers and stop, leaving you to trust them. This calculator substitutes each pair back into A·v = λv and reports the residual, so a correct answer carries its own proof and a wrong input is easy to spot.
Miss 3: no classification. The discriminant is computed silently elsewhere, if at all. Here the trace, determinant, discriminant and characteristic polynomial are all shown, with the three-case reading, so you can see which situation you are in and check the sum and product identities by hand.
One honest limit: this tool is deliberately a 2x2 specialist. Larger matrices need numerical methods rather than the quadratic formula, and repeated-eigenvalue matrices can be defective, with fewer independent eigenvectors than the size suggests. The calculator reports the repeated case when Δ = 0; deciding diagonalizability is a further step.
Frequently Asked Questions
A direction the matrix does not rotate: applying the matrix only stretches or flips vectors along it. The eigenvalue is the stretch factor, so the pair satisfies A·v = λv.
Real-coefficient polynomials have non-real roots in conjugate pairs, and the eigenvectors pair up the same way. If a + bi is an eigenvalue of a real matrix, a − bi is the other one.
No, by definition. The zero vector trivially satisfies A·0 = λ·0 for every λ, so it carries no information and is excluded. Eigenvectors are always nonzero.
No. Any nonzero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue. The tool normalizes real eigenvectors to unit length as a convention, so textbook answers that are multiples of the ones shown are equally correct.
Whether the eigenvalues are real and distinct (Δ > 0), repeated (Δ = 0), or a complex conjugate pair (Δ < 0). It is computed as trace² − 4·det.
One repeated eigenvalue. Most such matrices still have two independent eigenvectors, as scalar multiples of the identity do; a few defective matrices have only one, which makes them non-diagonalizable.
Yes. A real symmetric matrix, where the entry in row i, column j equals the entry in row j, column i, has all real eigenvalues and orthogonal eigenvectors, which is why symmetric matrices are so well behaved.
That is normal and expected: it means the transformation includes rotation. The rotation matrix [[0, −1], [1, 0]] is the classic example, with eigenvalues ±i and no real eigenvectors at all.
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References
- MIT OpenCourseWare, Mathematics 18.013A, "Computing Eigenvalues and Eigenvectors": the 2x2 characteristic equation x² − tr(M)x + det(M) = 0 and its solution by the quadratic formula. MIT OpenCourseWare: Computing Eigenvalues and Eigenvectors. Accessed October 2026.
- Khan Academy, "Introduction to eigenvalues and eigenvectors": the definition Av = λv and what eigenpairs mean for a transformation. Khan Academy: Eigenvalues and Eigenvectors. Accessed October 2026.
- Paul's Online Math Notes (Lamar University), "Eigenvalues and Eigenvectors": characteristic equation, eigenspaces, and worked 2x2 examples. Paul's Online Math Notes: Eigenvalues and Eigenvectors. Accessed October 2026.