Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools


Complex Eigenvalues



When Complex Eigenvalues Appear
Conjugate Pairs
The 2×2 Case in Detail
Complex Eigenvectors
Real Canonical Form
Complex Eigenvalues in Larger Matrices
Dynamical Systems Interpretation
The Fundamental Theorem of Algebra Connection
Summary: Complex Eigenvalues at a Glance



When No Real Direction Survives

A real matrix can have complex eigenvalues — this happens when the characteristic polynomial has no real roots. Geometrically, no real direction is mapped to a scalar multiple of itself; the transformation involves rotation. Complex eigenvalues of real matrices always come in conjugate pairs, and each pair corresponds to a rotation-scaling action on a two-dimensional subspace.



When Complex Eigenvalues Appear

A real matrix with real entries can have complex eigenvalues. This occurs when the characteristic polynomial — a polynomial with real coefficients — has roots that are not real.

For a 2×22 \times 2 matrix, complex eigenvalues appear when the discriminant tr(A)2−4det⁡(A)<0\text{tr}(A)^2 - 4\det(A) < 0. The quadratic formula produces λ=tr(A)±negative2\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{negative}}}{2}, which involves −1=i\sqrt{-1} = i.

The geometric interpretation is clear: no real direction in R2\mathbb{R}^2 is mapped to a scalar multiple of itself. Every vector is rotated, not just stretched or compressed. The simplest example is rotation by 90°90°, with matrix (0−110)\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} and characteristic polynomial λ2+1=0\lambda^2 + 1 = 0. The eigenvalues are ±i\pm i — purely imaginary, reflecting pure rotation with no scaling.

Conjugate Pairs

For a real matrix, complex eigenvalues always come in conjugate pairs. If λ=a+bi\lambda = a + bi is an eigenvalue (with b≠0b \neq 0), then λˉ=a−bi\bar{\lambda} = a - bi is also an eigenvalue.

The proof uses the fact that the characteristic polynomial has real coefficients. If p(λ)=0p(\lambda) = 0 and every coefficient of pp is real, then p(λˉ)=p(λ)‾=0‾=0p(\bar{\lambda}) = \overline{p(\lambda)} = \overline{0} = 0. Complex conjugation passes through the polynomial because the coefficients are their own conjugates.

The corresponding eigenvectors are also conjugates: if v\mathbf{v} is an eigenvector for λ=a+bi\lambda = a + bi, then vˉ\bar{\mathbf{v}} is an eigenvector for λˉ=a−bi\bar{\lambda} = a - bi.

One consequence for odd-dimensional real matrices: at least one eigenvalue must be real. Complex roots pair up, consuming an even number of the nn roots. When nn is odd, at least one root is left unpaired, and an unpaired root of a real polynomial must be real.

The 2×2 Case in Detail

A 2×22 \times 2 real matrix with complex eigenvalues λ=a±bi\lambda = a \pm bi (where b≠0b \neq 0) acts as a rotation composed with a scaling. The eigenvalues have modulus r=a2+b2r = \sqrt{a^2 + b^2} and argument θ=arctan⁡(b/a)\theta = \arctan(b/a).

When r=1r = 1 (equivalently, det⁡(A)=a2+b2=1\det(A) = a^2 + b^2 = 1), the transformation is a pure rotation by angle θ\theta. When r>1r > 1, it is a rotation with outward spiraling. When r<1r < 1, it is a rotation with inward spiraling.

Worked Example


For A=(3−211)A = \begin{pmatrix} 3 & -2 \\ 1 & 1 \end{pmatrix}: tr(A)=4\text{tr}(A) = 4, det⁡(A)=5\det(A) = 5, discriminant =16−20=−4= 16 - 20 = -4. The eigenvalues are

λ=4±−42=2±i\lambda = \frac{4 \pm \sqrt{-4}}{2} = 2 \pm i


The modulus is r=4+1=5r = \sqrt{4 + 1} = \sqrt{5} and the argument is θ=arctan⁡(1/2)\theta = \arctan(1/2). The transformation rotates by arctan⁡(1/2)≈26.6°\arctan(1/2) \approx 26.6° while scaling distances by 5≈2.24\sqrt{5} \approx 2.24. Since r>1r > 1, repeated application spirals outward.
Modulus r = |λ| Geometric action of A 2×2 marker (det A)
r = 1 pure rotation by θ; no scaling det(A) = a² + b² = 1
r > 1 rotation combined with outward scaling by r det(A) > 1
r < 1 rotation combined with inward scaling by r det(A) < 1

Complex Eigenvectors

The eigenvectors corresponding to complex eigenvalues have complex entries. To find them, solve (A−λI)v=0(A - \lambda I)\mathbf{v} = \mathbf{0} using complex arithmetic.

Continuing the example with A=(3−211)A = \begin{pmatrix} 3 & -2 \\ 1 & 1 \end{pmatrix} and λ=2+i\lambda = 2 + i:

A−(2+i)I=(1−i−21−1−i)A - (2 + i)I = \begin{pmatrix} 1 - i & -2 \\ 1 & -1 - i \end{pmatrix}


The second row gives v1+(−1−i)v2=0v_1 + (-1 - i)v_2 = 0, so v1=(1+i)v2v_1 = (1 + i)v_2. Setting v2=1v_2 = 1: v=(1+i,1)T\mathbf{v} = (1 + i, 1)^T.

This eigenvector splits into real and imaginary parts: v=(1,1)T⏟u+i(1,0)T⏟w\mathbf{v} = \underbrace{(1, 1)^T}_{\mathbf{u}} + i\underbrace{(1, 0)^T}_{\mathbf{w}}. The real vectors u\mathbf{u} and w\mathbf{w} encode the rotation — they span the two-dimensional subspace on which the rotation-scaling acts. The conjugate eigenvector for λˉ=2−i\bar{\lambda} = 2 - i is vˉ=(1−i,1)T\bar{\mathbf{v}} = (1 - i, 1)^T.

Real Canonical Form

A 2×22 \times 2 real matrix with eigenvalues a±bia \pm bi is similar (over R\mathbb{R}) to the matrix

(a−bba)=r(cos⁡θ−sin⁡θsin⁡θcos⁡θ)\begin{pmatrix} a & -b \\ b & a \end{pmatrix} = r\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}


where r=a2+b2r = \sqrt{a^2 + b^2} and θ=arctan⁡(b/a)\theta = \arctan(b/a). This is a rotation by θ\theta scaled by rr.

The similarity is achieved by the real matrix P=[u  w]P = [\mathbf{u} \; \mathbf{w}], where v=u+iw\mathbf{v} = \mathbf{u} + i\mathbf{w} is the complex eigenvector. Then P−1AP=(a−bba)P^{-1}AP = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}.

This is the real alternative to diagonalization: instead of a diagonal matrix with complex entries (which is valid over C\mathbb{C} but not over R\mathbb{R}), we get a real 2×22 \times 2 rotation-scaling block. The transformation is expressed in its simplest real form.

Complex Eigenvalues in Larger Matrices

For an n×nn \times n real matrix, complex eigenvalues still come in conjugate pairs. Each pair contributes a 2×22 \times 2 rotation-scaling block to the real canonical form. Real eigenvalues contribute 1×11 \times 1 blocks (just the eigenvalue itself).

A 4×44 \times 4 matrix with eigenvalues 2±3i2 \pm 3i and −1±i-1 \pm i has real canonical form

(2−300320000−1−1001−1)\begin{pmatrix} 2 & -3 & 0 & 0 \\ 3 & 2 & 0 & 0 \\ 0 & 0 & -1 & -1 \\ 0 & 0 & 1 & -1 \end{pmatrix}


A 3×33 \times 3 matrix with eigenvalues 55 and 1±2i1 \pm 2i has real canonical form

(50001−2021)\begin{pmatrix} 5 & 0 & 0 \\ 0 & 1 & -2 \\ 0 & 2 & 1 \end{pmatrix}


The real canonical form is the real analogue of diagonalization: it achieves the simplest possible real matrix that is similar to AA, with the block structure directly reflecting the eigenvalue spectrum.

Dynamical Systems Interpretation

Complex eigenvalues produce oscillatory behavior in dynamical systems.

In the discrete system xn+1=Axn\mathbf{x}_{n+1} = A\mathbf{x}_n, an eigenvalue λ=a+bi\lambda = a + bi with modulus r=∣λ∣=a2+b2r = |\lambda| = \sqrt{a^2 + b^2} generates a spiraling trajectory. When r>1r > 1, the spiral grows outward — the system is unstable. When r<1r < 1, the spiral decays inward — the system converges to the origin. When r=1r = 1, the trajectory traces a closed curve — perpetual oscillation without growth or decay.

In the continuous system x′=Ax\mathbf{x}' = A\mathbf{x}, an eigenvalue λ=a+bi\lambda = a + bi contributes a term eat(cos⁡bt,sin⁡bt)e^{at}(\cos bt, \sin bt) to the solution. The real part aa determines exponential growth (a>0a > 0) or decay (a<0a < 0). The imaginary part bb determines the oscillation frequency. When a=0a = 0, the oscillation is sustained. When a<0a < 0, it is damped. When a>0a > 0, it grows without bound.

Stability of a linear system reduces to eigenvalue analysis: the system is stable if and only if every eigenvalue has negative real part (continuous) or modulus less than 11 (discrete).
Regime Discrete: xn+1 = A xn Continuous: x′ = A x
Stable (converging) |λ| < 1 → trajectory spirals inward to the origin Re(λ) < 0 → damped oscillation decaying to zero
Marginally stable |λ| = 1 → trajectory traces a closed (periodic) curve Re(λ) = 0 → sustained oscillation at constant amplitude
Unstable (diverging) |λ| > 1 → trajectory spirals outward without bound Re(λ) > 0 → oscillation with exponentially growing amplitude

The Fundamental Theorem of Algebra Connection

Over C\mathbb{C}, every polynomial of degree nn factors completely into nn linear factors. This means every n×nn \times n matrix — real or complex — has exactly nn eigenvalues counted with algebraic multiplicity when the characteristic polynomial is factored over C\mathbb{C}.

Over R\mathbb{R}, some factors may be irreducible quadratics rather than linear, corresponding to conjugate pairs of complex eigenvalues. The characteristic polynomial of a real matrix factors into linear terms (real eigenvalues) and irreducible quadratic terms (complex conjugate pairs).

Working over C\mathbb{C} simplifies the theory considerably. Every matrix is triangularizable over C\mathbb{C} (Schur decomposition): A=UTU∗A = UTU^* where UU is unitary and TT is upper triangular with eigenvalues on the diagonal. Diagonalizability depends only on whether the geometric multiplicities match the algebraic multiplicities, with no additional complications from irreducible quadratics.

The choice between working over R\mathbb{R} and C\mathbb{C} is a recurring theme. Real matrices are the natural objects of computation, but complex eigenvalues are the natural objects of spectral theory. Both perspectives are needed.

Summary: Complex Eigenvalues at a Glance

The behavior of complex eigenvalues for a real matrix is governed by a handful of formulas and statements that connect the discriminant, the modulus, the argument, and the real part to specific geometric and dynamical consequences. The table below collects each one alongside what it produces and what it means, providing a single reference for the page&apos;s key facts.
Aspect of complex eigenvalues λ = a ± bi Formula or statement Meaning
When they appear (2×2) discriminant tr(A)² − 4·det(A) < 0 no real direction is mapped to a scalar multiple of itself
Pair structure if λ = a + bi is an eigenvalue, so is λ̄ = a − bi eigenvectors come in conjugate pairs v and v̄
Modulus r r = √(a² + b²) scaling factor of the rotation-scaling action
Argument θ θ = arctan(b / a) rotation angle of the action
Real canonical 2×2 block [[a, −b], [b, a]] = r · R(θ) A is similar over ℝ to a rotation by θ scaled by r
Discrete stability stable ⟺ |λ| < 1 for every eigenvalue criterion for xn+1 = A xn
Continuous stability stable ⟺ Re(λ) < 0 for every eigenvalue criterion for x′ = A x
Odd-dimensional real matrix at least one eigenvalue is real complex roots come in pairs, so an unpaired root must be real