Generalised eigenvalue problem — where it appears
Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.
A spectrum that comes in reciprocal pairs
A palindromic quadratic reads the same backwards, so λ is an eigenvalue exactly when 1/λ is. A general solver discards that, computes the large half of the spectrum perfectly and the small half to seven digits — and the small half is a division away from being perfect too.
An augmentation read in the smallest eigenvalue
An augmented Lagrangian preconditioner's least work sat at γ = 1, 10⁴ and 1 on three systems, and two of the three optima sat where the smallest generalised eigenvalue was about 0.06 — which suggested a rule written in ν rather than γ. On twenty-four systems whose norms are all one, the least-work γ spans seven decades and ν at the optimum spans a factor of thirty, so there is no one ν. There is still a rule: take the smallest γ at which ν reaches 0.03, and the median system pays 7 per cent over its least work and the worst 29, where the best single γ pays 61. Its price is digits — twelve times the best forward error on the median system, 4,561 times on the worst.
The rank that stops being typical
A random 2 × 2 × 2 tensor has rank two with probability π/4 and rank three otherwise, and the sentence has no analogue for matrices. It is the first of a family. An n × n × 2 tensor is a pencil of two slices, and it has rank n exactly when the pencil's eigenvalues are all real, n + 1 otherwise. Over draws, the share of rank n is 0.786 at n = 2, 0.500 at 3, 0.264 at 4, 0.039 at 6, 0.004 at 8 and none of 4,000 at 10 — falling like e^(−0.087n²) — because the mean number of real eigenvalues grows only like the square root of n, to Edelman, Kostlan and Shub's closed form within two per cent. Both ranks stay typical in theory; in practice the lower one disappears.
A fit with no answer to find
Half of all random 3 × 3 × 2 tensors, and most larger ones, have no rank-three decomposition, because their pencil has a complex pair. A rank-n fit to one of them does not wander and does not stall. Two starts settle at the same error to five digits, and that error is the distance from the tensor to the surface where its pencil has a double eigenvalue — found with no fitting at all, and matched to within one and a half per cent. Meanwhile the fit's terms grow without limit, like the square root of the sweep count, while the fitted pencil's two closest eigenvalues close on each other at exactly the rate the terms grow. The error has an answer; the decomposition does not.
Two matrices and one problem
Ax = λBx is what a finite element model, a structural vibration and a constrained optimisation actually produce, and it is not the one-matrix problem with a change of variables. Everybody is told not to form B⁻¹A because it is not symmetric. That is true, the departure from symmetry is about one, and it is not what decides the accuracy.
An eigenvalue with no value
If the second matrix of a pencil is singular then some of the eigenvalues are infinite, and that is not a degeneracy — it is the algebraic constraints of the model, one per constraint. What survives is a pair of numbers rather than one, and on the line those pairs live on, infinity is an ordinary point with an ordinary residual.
A problem with no answer
If two matrices share a null vector then det(A − λB) is identically zero and every λ is an eigenvalue, which means none of them is. Perturb such a pencil by a ten-billionth and a solver returns six numbers with residuals below 10⁻⁹. Change the seed and it returns six different numbers, spread over forty-four, with residuals just as small.
Named alongside it
The objects these essays reach for when they reach for this one.
Matrix pencilBackward errorBorder-rankCP decompositionDeterminantEigenvaluesExact arithmeticTensor rankAlternating least-squaresAugmented lagrangianBlock preconditionerCayley transform