Complex Eigenvalue Calculator
Some matrices stretch space; others twist it. When a 2×2 real matrix rotates as it scales, its eigenvalues leave the real line and arrive as a complex conjugate pair, a ± bi, and ordinary real-only calculators either refuse them or round them into nonsense. This calculator solves 2×2 matrices exactly, real or complex, solves 3×3 matrices through their characteristic cubic, and shows the characteristic polynomial it solved so you can check every number.
Scope, stated plainly: 2×2 matrices get exact eigenvalues (real, repeated, or a complex pair) plus the characteristic polynomial. 3×3 matrices get all three eigenvalues from their characteristic cubic, solved in closed form as described below, plus the cubic itself. Eigenvectors are not computed. Entries are real numbers only.
Quick Answer
Key takeaways
- For a 2x2 matrix, everything flows from two numbers: the trace (a + d) and the determinant (ad − bc).
- The discriminant, trace² − 4 x determinant, decides the case before any roots are computed: positive gives two real eigenvalues, zero gives a repeated one, negative gives a complex conjugate pair.
- Complex eigenvalues are not exotic: any real matrix with a rotational character, from a 90-degree rotation to a damped oscillator model, produces them.
- For a 3x3 matrix, three numbers play the trace-and-determinant role: the trace, the sum of the three principal 2x2 minors, and the determinant. Together they build the characteristic cubic, and its three roots are the eigenvalues.
- This page is honest about its edges: no eigenvectors and no complex entries, in either size.
- Quick Answer
- How to use the complex eigenvalue calculator
- The trace and determinant shortcut
- Worked examples: all three cases
- What complex eigenvalues mean in real systems
- How the 3x3 mode works: the characteristic cubic
- What competing calculators miss
- Frequently Asked Questions
- Add This Calculator to Your Website
- References
How to use the complex eigenvalue calculator
Leave the size on 2x2, type the four entries in reading order (top row a11, a12; bottom row a21, a22), and press Calculate Eigenvalues. Blank or non-numeric entries are treated as zero, matching how the matrix would read with a 0 in that seat. The result shows both eigenvalues to four decimal places and the characteristic polynomial beneath them, printed with the trace and determinant uncombined so you can see exactly which numbers produced the roots.
The same steps work for a 3x3 matrix: switch the size selector to 3x3, fill the nine entries, and press Calculate Eigenvalues. The grid reloads with a pre-filled 3x3 example, and the result gains a third eigenvalue row plus the characteristic cubic beneath it. The method behind the 3x3 mode, and a fully worked 3x3 example, are in the 3x3 section below.
Read a complex result as a pair, never as two separate answers: 0.0000 + 1.0000i and 0.0000 − 1.0000i together describe one rotational behaviour. If you go on to solve for the vectors by hand, the polynomial is also your starting point for the eigenspace work that a basis for eigenspace calculator formalises, and each eigenvector you find must sit in the null space you can check with the basis for null space calculator.
The trace and determinant shortcut
For 2x2 matrices there is no need to expand a determinant with λ in it and then wrestle the quadratic. The characteristic polynomial is always the same shape, built from the trace and the determinant, as the LibreTexts characteristic polynomial treatment shows.
Two free checks come with the method. The eigenvalues always add up to the trace, and they always multiply to the determinant, real or complex. For the worked example: 2 + (−1) = 1 = trace, and 2 x (−1) = −2 = determinant. For the rotation pair: i + (−i) = 0 = trace, and i x (−i) = 1 = determinant. If your hand calculation fails either check, the error is in the arithmetic, not the method, a self-audit the MIT OpenCourseWare eigenvalue problem sessions lean on heavily.
Worked examples: all three cases
Every 2x2 real matrix lands in exactly one of three cases, decided by the sign of the discriminant. All three rows below are outputs of this calculator, verified against hand calculation.
| Matrix | T, D, discriminant | Eigenvalues | Case |
|---|---|---|---|
| [[4, -5], [2, -3]] | 1, -2, 9 | 2.0000 and -1.0000 | Two real |
| [[0, -1], [1, 0]] | 0, 1, -4 | 0.0000 + 1.0000i and 0.0000 - 1.0000i | Complex pair |
| [[2, 0], [0, 2]] | 4, 4, 0 | 2.0000 and 2.0000 | Repeated |
The rotation row deserves a second look. [[0, −1], [1, 0]] rotates every vector by 90 degrees, so no real direction survives the transformation unchanged, which is precisely why no real eigenvalue exists: an eigenvector would have to point the same way after being turned a quarter circle. The eigenvalues ±i encode that: modulus 1 (no scaling, a pure rotation), argument 90 degrees (the rotation itself). Matrices that spiral, such as [[1, −1], [1, 1]], give pairs like 1 ± i, where the real part 1 is the growth centre and the imaginary part is the turn.
What complex eigenvalues mean in real systems
A complex pair is the signature of oscillation. In a system of differential equations, a pair a ± bi produces solutions built from e^(at) multiplied by sines and cosines of bt: the real part sets growth or decay, the imaginary part sets the frequency. Negative a means the oscillation dies away (a stable spiral), positive a means it grows, and a = 0 means a pure, sustained cycle like the rotation example. The same pairs appear in control systems, AC circuit analysis, vibration engineering, and quantum mechanics, anywhere a system's natural behaviour is to cycle rather than settle. The LibreTexts complex eigenvalues section develops the rotation-scaling picture behind this: a 2x2 matrix with a complex pair acts, in the right basis, as a rotation composed with a uniform scaling.
How the 3x3 mode works: the characteristic cubic
A 3x3 characteristic polynomial is a cubic, λ³ − Tλ² + Mλ − D = 0, built from three numbers: the trace T (a11 + a22 + a33), the sum M of the three principal 2x2 minors (the 2x2 determinants you get by keeping each pair of rows and the same pair of columns), and the determinant D. This tool builds that cubic, then solves it in closed form. Shifting λ by T/3 removes the squared term and leaves a depressed cubic, y³ + py + q = 0. One discriminant, (q/2)² + (p/3)³, then decides the case, exactly as the 2x2 discriminant does. When it is negative, all three roots are real and the trigonometric form of Cardano's method returns them. When it is zero, at least two roots repeat, and the same trigonometric form returns the repeated value twice. When it is positive, Cardano's cube-root formula returns the one real root, and the other two roots form a complex conjugate pair whose real and imaginary parts are built from the same two cube roots.
Worked example, using the matrix the 3x3 grid loads with: A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. The trace is 1 + 1 + 0 = 2. The principal minors are (1 x 1 − 2 x 0) = 1, (1 x 0 − 3 x 5) = −15, and (1 x 0 − 4 x 6) = −24, so M = 1 − 15 − 24 = −38. The determinant is 1. The cubic is therefore λ³ − 2λ² − 38λ − 1 = 0, and this calculator returns its roots as 7.2560, −0.0264 and −5.2297. The checks pass on the unrounded roots (7.256022, −0.026353, −5.229670): they sum to 2, the trace; their pairwise products sum to −38, the minor sum; and their product is 1, the determinant. Each root also makes the cubic itself zero, to within floating-point rounding. The same roots come out of the LAPACK routines behind NumPy's eig function, which is what these worked values were checked against.
A 3x3 matrix can also produce a complex pair. The block rotation [[0, −1, 0], [1, 0, 0], [0, 0, 2]] rotates the xy-plane while stretching the z-axis by 2, and the tool returns 2.0000 alongside 0.0000 + 1.0000i and 0.0000 − 1.0000i: the real eigenvalue for the axis that survives, and the conjugate pair for the plane that only turns. Repeated eigenvalues are reported with their multiplicity, so the identity matrix returns 1.0000 three times, and a matrix with a double eigenvalue returns that value twice.
What competing calculators miss
The field for this keyword is strong, and honesty requires saying so: step-by-step sites such as eMathHelp and full suites such as MiniWebtool return eigenvectors and worked steps that this page does not. What this calculator does differently is narrower.
Its edge: the complex pair is a first-class result. Many quick 2x2 tools are built for the real case and report a negative discriminant as an error or "no real eigenvalues", which is exactly the dead end this page exists to avoid. Here the pair is formatted, signed, and explained, and the characteristic polynomial is shown uncombined so the discriminant's origin is visible.
Miss in the other direction: no eigenvectors, no steps. If your course marks the working, you will still need the (A − λI)v = 0 solve by hand; this tool verifies the λ values you should land on and nothing more. Treat it as the fast check at the end of a hand calculation, or the front end that tells you which case you are in before you commit to a method. A second fast check lives at the eigen values calculator, which finds eigenvalues for 2x2 and 3x3 matrices and shows the characteristic polynomial behind each result.
Frequently Asked Questions
An eigenvalue with a non-zero imaginary part, written a + bi. For a real matrix it means the transformation combines rotation with scaling rather than pure stretching, which is why complex eigenvalues govern oscillation and spiral behaviour in dynamic systems.
A real matrix has a characteristic polynomial with real coefficients, and non-real roots of a real polynomial always arrive as a + bi together with a - bi. So a 2x2 real matrix has either two real eigenvalues, one repeated real eigenvalue, or one conjugate pair.
No. It returns the eigenvalues and the characteristic polynomial for 2x2 and 3x3 matrices. Eigenvectors need a separate solve of (A - lambda I)v = 0 for each eigenvalue; the characteristic polynomial shown here is the starting point for that work by hand. That follow-on solve has its own tool: the complex eigenvector calculator returns both eigenvalues and their eigenvectors for a 2x2 matrix, including the complex pair case, and checks each pair against the matrix.
It computes all three eigenvalues and shows the characteristic cubic it solved. The cubic is built from the trace, the sum of the three principal 2x2 minors, and the determinant, and it is solved in closed form: the trigonometric form of Cardano's method when all three roots are real, and Cardano's cube-root formula when one real root sits next to a complex conjugate pair. Results are shown to four decimal places, and a repeated eigenvalue is printed once per repetition.
No. Entries are read as real numbers, so text like 2+3i would be read as just 2 and silently give wrong eigenvalues. The complex numbers here are outputs, produced when a real matrix has a negative discriminant, not inputs.
For a 2x2 matrix it is lambda squared minus trace times lambda plus determinant. Its roots are the eigenvalues, so the calculator shows it as the audit trail for the two numbers above it. The tool prints the trace and determinant uncombined, so a trace of 1 and determinant of -2 appears as lambda^2 - 1 lambda + -2, which reads as lambda^2 - lambda - 2. For a 3x3 matrix the polynomial is a cubic, lambda cubed minus trace times lambda squared plus the principal-minor sum times lambda minus determinant, and the tool prints it in the same uncombined style.
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References
- Mathematics LibreTexts, "Characteristic Polynomial" - the theorem that eigenvalues are exactly the roots of det(A - lambda I), with the 2x2 trace and determinant form. LibreTexts: Characteristic Polynomial. Accessed October 2026.
- Mathematics LibreTexts (UC Irvine Math 3A), "Complex Eigenvalues and Eigenvectors" - conjugate pairs, and why a matrix with complex eigenvalues acts as a rotation-scaling. LibreTexts: Complex Eigenvalues. Accessed October 2026.
- MIT OpenCourseWare, "Problem Solving: Eigenvalues and Eigenvectors" (18.06SC Linear Algebra) - worked eigenvalue problems and the trace and determinant checks. MIT OCW: Eigenvalue problem solving. Accessed October 2026.
- Mathematics LibreTexts (Hartman, Fundamentals of Matrix Algebra), "Eigenvalues and Eigenvectors" - the eigenvalue equation, eigenspaces, and solving (A - lambda I)v = 0. LibreTexts: Eigenvalues and Eigenvectors. Accessed October 2026.
- NumPy Developers, "numpy.linalg.eig" - eigenvalues computed with the LAPACK _geev routines; eigenvalues of a real matrix are real or arrive in conjugate pairs. The 3x3 worked values and the solver verification battery on this page were checked against it. NumPy: numpy.linalg.eig. Accessed October 2026.