Eigenvalues and eigenvectors

The formula

A·v = λ·v — apply the matrix A to the vector v and you get the same vector back, scaled by λ. That v is an eigenvector; that λ is its eigenvalue.

det(A − λ·I) = 0 — the characteristic equation. Its roots are the eigenvalues. An n×n matrix gives a degree-n polynomial, so n eigenvalues, counting repeats.

K·φ = ω²·M·φ — the generalized eigenvalue problem for structures. K is the stiffness matrix, M is the mass matrix, each ω is a natural frequency in rad/s, and each φ is the mode shape that vibrates at it.

λ = σ ± i·ω_d — for a state matrix in ẋ = A·x, a complex eigenvalue pair reads as a damped oscillation: σ is the decay rate, ω_d is the ringing frequency. Negative σ, stable. Positive σ, you have a problem.

Where you meet it

The modal survey. A shaker, a grid of accelerometers, and an article on the test stand. The peaks in the frequency response are the structure's natural frequencies — the eigenvalues of K·φ = ω²·M·φ — and the deflection patterns at those peaks are the eigenvectors. Model correlation means the FEA's eigenvalues and the test's peaks agree within a few percent, or somebody updates the model.

The control design review. The board asks where the closed-loop poles sit. Poles of a linear system are eigenvalues of its A matrix. Every real part negative, the loop is stable; one eigenvalue crosses into the right half-plane and the actuator finds out first.

The stress rollup. The 3×3 stress tensor at a point has three eigenvalues — the principal stresses — and its eigenvectors are the planes they act on. Plane-stress example: σx = 80 MPa, σy = 20 MPa, τxy = 30 MPa gives principal stresses of 92.4 and 7.6 MPa. That 92.4 is what goes against the allowable, not the 80 you measured on the gauge axis.

The stability check. Euler buckling loads and shaft critical speeds are eigenvalues too — the smallest eigenvalue of a stability problem is the first load or speed where the structure has a second equilibrium shape available. You care about the smallest one for the same reason you care about the first mode.

How it works

Think of a matrix as a machine that moves vectors. Feed most directions in and they come out rotated somewhere else. A few special directions come out pointing exactly where they went in, just longer or shorter. Take A = [[2, 1], [1, 2]]: the direction [1, 1] maps to [3, 3] — same direction, tripled, so λ = 3. The direction [1, −1] maps to itself untouched, so λ = 1. Every other input direction turns.

Three behaviors worth having in your head:

Eigenvectors are directions, not vectors. Any scalar multiple of an eigenvector is the same eigenvector. That's why mode shapes come out of a solver with arbitrary amplitude and get mass-normalized or scaled to unit maximum before anyone compares them.
Symmetric matrices are the friendly case. A real symmetric matrix — stress tensors, inertia tensors, K and M from a proper FE model — always has real eigenvalues and mutually orthogonal eigenvectors. That orthogonality is the entire foundation of modal analysis: it's what lets you decouple a thousand-DOF structure into independent single-DOF oscillators, one per mode. It's also why principal stresses always exist, for any load state.
Non-symmetric matrices can go complex. Losing symmetry removes the guarantee of real eigenvalues — it doesn't force complex ones. Plenty of non-symmetric matrices stay all-real: any triangular matrix, an overdamped system, a chain of first-order lags. But when a real matrix does produce complex eigenvalues, they arrive in conjugate pairs, and each pair reads as a damped oscillation — a frequency and a decay rate together — while a real eigenvalue stays a pure exponential decay or growth, no ringing. The extreme case is a pure 90° rotation matrix, with eigenvalues ±i: no real direction survives a rotation, so nothing real is left for the eigenvalues to be.

The gotchas. A repeated eigenvalue can be short on eigenvectors — [[1, 1], [0, 1]] has λ = 1 twice but only one eigenvector direction. Such "defective" matrices produce responses that grow like t·e^(λt), which is why marginal cases in stability analysis deserve suspicion rather than benefit of the doubt. Separately: nearly-defective matrices have eigenvalues that are violently sensitive to small perturbations, so if two computed poles sit almost on top of each other, treat their exact values as soft numbers.

The mistake people make: computing eigenvalues by expanding det(A − λ·I) and root-finding the polynomial. That's the correct definition and the correct hand method for a 2×2 on paper. In floating point at any real size it's a disaster — polynomial roots are far more sensitive to coefficient error than the eigenvalues are to matrix error, so the detour through the characteristic polynomial manufactures error out of nothing. Every serious library (LAPACK underneath MATLAB's and NumPy's eig) goes the other way: orthogonal similarity transforms and QR iteration, never the polynomial. If your FEA or your control toolbox gives you eigenvalues, it earned them that way.

One more limit of validity: an FE model's eigenvalues are only as trustworthy as the mesh and the boundary conditions. The first few modes of a decent model are usually solid; mode 40 is fiction until test says otherwise. Fixity assumptions move frequencies more than most people expect — a "clamped" base that's really three bolts into a plate will read low on the stand every time.

History

The problem showed up in mechanics long before anyone called it an eigenvalue problem. Euler, working out the mechanics of spinning rigid bodies in his 1765 Theoria motus corporum solidorum seu rigidorum, found that every body has three principal axes about which it rotates cleanly — and coined "moment of inertia" while he was at it ^[1]^[2]. Lagrange later recognized those principal axes as exactly the eigenvectors of the inertia matrix ^[1]^[3].

The algebra caught up in 1826, when Cauchy worked out eigenvalues and diagonalization while converting quadratic forms to sums of squares, and named the roots of det(A − λ·I) = 0 the racines caractéristiques — characteristic roots, the term engineers' textbooks kept for the next century ^[1]^[4]. Working from ideas Sturm brought to him out of Fourier's heat-equation work, Cauchy also established the result that makes structural dynamics behave: real symmetric matrices have real eigenvalues ^[1]^[4].

The word itself came from Göttingen. In 1904, David Hilbert, studying integral equations as infinite-dimensional versions of the same problem, started writing Eigenfunktion and Eigenwert — "own function," "own value" ^[1]^[3]. The half-translated hybrid "eigenvalue" eventually displaced "characteristic root" in English, over objections from people who thought half-translating a German word was worse than not translating it at all.

Computation is the modern chapter. Richard von Mises and Hilda Pollaczek-Geiringer published the power method in 1929 — multiply a trial vector by A repeatedly and it collapses onto the dominant eigenvector ^[5]^[6]. Then, around 1959–1961, John Francis in London and Vera Kublanovskaya in Leningrad independently developed the QR algorithm ^[1]^[7]^[8]. It ranks among the most consequential numerical methods of the twentieth century, and it is almost certainly what ran the last time you typed eig.

Related tools

/tools/vibration-natural-freq — the 1-DOF special case; a full modal analysis is this calculator repeated once per eigenvalue
/tools/shaft-speed-critical — critical speed is the first eigenvalue of the rotor's bending problem
/tools/euler-buckling — the buckling load is the smallest eigenvalue of a column's stability problem
/tools/moment-of-inertia-shapes — principal axes of a section are eigenvectors of its inertia tensor
/tools/stress-strain — principal stresses, the eigenvalues every margin writeup actually uses
/tools/lc-resonance — resonant frequency of an LC tank, the circuit-side cousin of a structural mode

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →