Eigenvalues and eigenvectors
In every transformation there are privileged directions that do not turn, they only stretch. Finding them means finding the natural modes of a physical system.
01Directions that do not turn
A linear transformation, in general, moves every vector and changes its direction. But there are almost always a few privileged directions along which the vector does not turn: it is only stretched, shortened or flipped. Those directions are the eigenvectors, and the factor by which they are stretched, the eigenvalues.
They are also called proper (or characteristic) values and vectors. The prefix comes from the German eigen, which means “own”: they are characteristic of the transformation, not of the coordinate system used to write it.
02How they are computed
The condition A·v = λ·v is rewritten as (A − λI)·v = 0, which is a homogeneous system. That system has solutions other than the trivial one only if its determinant vanishes:
For 2 × 2 it is especially compact, using the trace (the sum of the diagonal) and the determinant:
Once each λ is found, the eigenvector is obtained by solving the homogeneous system (A − λI)·v = 0. The solution is never unique: if v is an eigenvector, any multiple of it is one too. What matters is the direction, not the length.
A = [[4, 1], [2, 3]]: tr = 7 and det = 12 − 2 = 10, so λ² − 7λ + 10 = 0 and the eigenvalues are λ₁ = 5 and λ₂ = 2.
For λ = 5: (A − 5I) = [[−1, 1], [2, −2]], and the equation −x + y = 0 gives the direction v₁ = (1, 1). Check: A·(1,1) = (5, 5) = 5·(1,1) ✔
For λ = 2: [[2, 1], [2, 1]], and 2x + y = 0 gives v₂ = (1, −2). Check: A·(1,−2) = (2, −4) = 2·(1,−2) ✔
The sum of the eigenvalues is the trace and their product is the determinant. In the example: 5 + 2 = 7 ✔ and 5 · 2 = 10 ✔. If they do not match, there is an arithmetic error to fix before going on.
03Seeing them
The circle of unit vectors is transformed into an ellipse. The directions where the blue and the orange arrows are aligned are the eigenvectors, and the ratio of lengths is the eigenvalue.
04Diagonalization
If a matrix of order n has n linearly independent eigenvectors, they can be used as a basis. In that basis the transformation only stretches each axis: its matrix is diagonal, with the eigenvalues on the diagonal.
- If the n eigenvalues are distinct, the matrix is always diagonalizable.
- If any of them is repeated, it may not be: the shear [[1, 1], [0, 1]] has a double λ = 1 and only one eigen-direction.
- Every symmetric matrix with real entries is diagonalizable, its eigenvalues are real and its eigenvectors are perpendicular to each other. This is the theorem behind the reduction to principal axes in quadratic forms.
- The eigenvalues can be complex, and then no real direction is preserved: this is what happens with a rotation, and in a circuit it means oscillation.
05What they represent in a physical system
A circuit with inductors and capacitors is described by a system of first-order differential equations, written in matrix form as ẋ = A·x, where x gathers the inductor currents and the capacitor voltages. The solution is built from exponentials eλt, and those λ are the eigenvalues of A: the natural modes of the circuit, which are the same poles that appear in the frequency response.
| Eigenvalues of A | Behavior | Example |
|---|---|---|
| Negative real | Exponential decay without oscillating: stable, overdamped | RC with large resistance |
| Complex with negative real part | Oscillation that dies out: stable, underdamped | RLC with little damping |
| Purely imaginary | Oscillation that does not die out | Ideal lossless LC |
| With positive real part | Grows without bound: unstable | Poorly compensated feedback loop; an oscillator at start-up |
Hence the stability criterion is “all eigenvalues with negative real part”. The same idea reappears in control, in the resonant frequencies of a mechanical structure, in the modes of a waveguide and in the principal components of a data set.
06In the lab
Compute the eigenvalues and eigenvectors of [[2, 1], [1, 2]] and of [[0, 1], [−2, −3]]. Verify in each case with the trace and the determinant, and check A·v = λ·v by substitution. Compare with the lab.
Write ẋ = A·x for a series RLC circuit with x = (iL, vC), compute the eigenvalues for three values of R —overdamped, critical and underdamped— and compare with the waveform obtained in a simulation.
Take any vector and multiply it repeatedly by A, normalizing each time. Observe that it aligns with the eigenvector of the eigenvalue of largest magnitude. It is the “Iterate” button of the lab, and it is the algorithm with which Google used to rank its results.
07Common mistakes
- Looking for the eigenvector before the eigenvalue. First solve det(A − λI) = 0.
- Subtracting λ only from the first element of the diagonal instead of from the whole diagonal.
- Giving the zero vector as an eigenvector: it is excluded by definition.
- Believing the eigenvector is unique: the whole line it spans is.
- Assuming that every matrix is diagonalizable. With repeated eigenvalues you have to check how many independent eigenvectors there are.
- Discarding complex eigenvalues: they are precisely the ones that describe oscillations.
08Self-assessment
Eigenvalues of [[3, 0], [0, −2]].
It is diagonal: 3 and −2, with eigenvectors î and ĵ.
If tr(A) = 6 and det(A) = 8 for a 2 × 2 matrix, what are its eigenvalues?
λ² − 6λ + 8 = 0 → λ = 4 and λ = 2.
What does a zero eigenvalue mean?
That the matrix is singular: there is a direction that ends up at the origin, the kernel is not trivial and det(A) = 0.
Why does a 90° rotation in the plane have no real eigenvectors?
Because no direction is preserved: every vector turns. Its eigenvalues are ±i, purely imaginary, which is the signature of a pure oscillation.
A circuit has eigenvalues −100 ± j2000 s−1. How does it respond?
It oscillates at about 2000 rad/s (318 Hz) with an envelope that decays with a time constant of 1/100 = 10 ms: it is stable and underdamped.
What is diagonalizing good for?
For working in the basis where the transformation only stretches each axis: powers, matrix exponentials and the solution of systems of differential equations become immediate.
09Further reading
- Stanley I. Grossman. Álgebra lineal. 6th ed., McGraw-Hill, 2008 (in Spanish). Eigenvalues, diagonalization and applications to systems of differential equations.
- Gilbert Strang. Linear Algebra and Its Applications. 4th ed. (Spanish edition, “Álgebra lineal y sus aplicaciones”), Cengage Learning, 2007. Chapter 5 connects eigenvalues with stability, powers and the matrix exponential.
- Norman S. Nise. Control Systems Engineering. 3rd ed. (Spanish edition, “Sistemas de control para ingeniería”), CECSA, 2006. Where they are used in practice: state variables, poles and stability of feedback systems.