Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Descriptor system — where it appears

A linear system written as E x' = A x + B u with E possibly singular, so its transfer function is C(sE - A)^-1 B. The Loewner realisation returns one, and projecting it to the right order is what makes the pencil regular.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

12345678910111210⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10¹order of the reduced modelworst relative errora model made of measurementssamples used24degree read6gap at the cut2·10⁸best model, order12its error2.7·10⁻¹¹states in the original40filled: at its own samplesopen: everywhere else

A model with no matrices behind it

Twenty-four numbers — values of a transfer function at twenty-four points — produce a sixth-order model of a forty-state system that passes through every sample to 10⁻¹² and matches the function it never saw to 10⁻⁹. Noise of 10⁻¹⁰ on those numbers takes the rank decision's gap from 10⁸ to 4.

reduction · Data-driven realisation finite eigenvalues (degree of det Q)13at infinity (2n − degree)1at infinity, by the rank of M12n, if M were nonsingular14a degree, not a decisiondegree of det Q13at infinity1by the rank of M1singular-value gap∞the count is a degreeand the other route is a judgement

One mass removed, and one eigenvalue gone

A coordinate with no inertia reads like a coordinate that has been deleted, and a chain of eight masses with one of them removed would then be a chain of seven, with fourteen eigenvalues. It has fifteen. The massless coordinate is still there, still carrying a damper, and it contributes a first-order equation rather than none.

polynomial · Polynomial eigenvalue λ = 0λ = ∞λ = ∞-0.37690.5654.3836.4282 eigenvalues here, and it is one placecounted exactly, in rationalsfinite eigenvalues4at infinity2degree of det(A − λB)4worst residual, either kind5.5·10⁻¹⁴an eigenvalue is a ratioand a ratio has a direction, not a size

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.

spectra · Pencil σ14.87σ22.55σ31.08σ40.782σ58.04·10⁻¹⁷σ610⁻¹⁸σ710⁻¹⁸n = 7, and det(A − λB) has degree 4an integer, and a judgementdegree of det(A − λB), exactly4infinite eigenvalues, from the degree3singular values below the cut3largest gap in the spectrum∞a degree cannot be nearly threeand a singular value can be nearly zero

The largest gap is inside the null space

The rule recommended for counting a pencil's infinite eigenvalues is to cut at the largest gap in the singular values of B. On integer pencils, with no perturbation anywhere and an exact answer available from the characteristic polynomial, it returns the wrong count on nine of twenty-five — because the singular values that are mathematically zero come back spread over a hundred and forty orders of magnitude, and the largest ratio in the list is between two of them.

spectra · Pencil

Named alongside it

The objects these essays reach for when they reach for this one.

DeterminantInfinite eigenvalueExact arithmeticMatrix pencilNumerical rankSingular valuesAdaptive interpolationCharacteristic polynomialData-driven realisationExact ground truthGeneralised eigenvalue problemInterpolation

All concepts