Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Generalised eigenvalue problem — where it appears

Ax = λBx, the form a finite element model, a structural vibration and a constrained optimisation produce rather than Ax = λx.

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

Also named here as matrix pencil — the same set of essays touches all of them, so they are one junction rather than several.

10¹10⁴10⁷10¹⁰10¹³10¹⁶10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹κ(B)relative error, and asymmetryvia B⁻¹Avia Choleskyasymmetry of B⁻¹Au · κ(B)against a spectrum known exactlyslope, via B⁻¹A0.92slope, via Cholesky0.98worst ratio between them2.3asymmetry of B⁻¹A1.1the symmetry claim is trueand it is not about the accuracy

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.

spectra · pencil 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 -40-27-14-11225380123456789computed eigenvalueseedevery mark has a residual below 10⁻⁸a small residual, and no answercoefficients of det(A − λB)0worst residual over all seeds1.8·10⁻⁹spread of the answers66seeds drawn8the residual is small at every markand none of the marks means anything

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.

spectra · pencil

Named alongside it

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

Matrix pencilBackward errorDeterminantExact arithmeticCholeskyCondition numberCounterexampleDescriptor systemEigenvalue condition numberEigenvaluesIll posed problemInfinite eigenvalue

All concepts