Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

The real Schur form — where it appears

The quasi-triangular matrix an orthogonal similarity reduces a real matrix to, with 2×2 blocks where a complex conjugate pair would otherwise appear. It is what a real eigensolver actually returns, and its 2×2 blocks are how a complex conjugate pair is represented without leaving real arithmetic.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

3000000120000-21000000-0.5-1.500001.5-0.5000000-2T = ZᵀAZthe highlighted boxes each hold one conjugate pair, and no real rotation removes themthe form, and that it is one‖A − ZTZᵀ‖/‖A‖1.8·10⁻¹⁵‖ZᵀZ − I‖2.5·10⁻¹⁵worst eigenvalue error2.7·10⁻¹⁵surviving subdiagonal26×6, spectrum chosen before the matrix was builtquasi-triangular is as far as the reals go

The form a real matrix can reach

A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.

spectra · Real schur darker is smaller: 10⁻¹⁰, 10⁻⁶, 10⁻³, 10⁻¹one spectrum, two matricesspectral radius0.8reach of the 10⁻³ level1.3eigenvalues, all at0.8the circle is |z| = 1and every eigenvalue is well inside it

The eigenvalues that are not there

For a normal matrix the resolvent norm is exactly one over the distance to the nearest eigenvalue, so a picture of it carries nothing the spectrum did not. Move one entry above the diagonal and the region a perturbation of 10⁻⁸ can put an eigenvalue into stops being a disc and reaches out past the unit circle, while every eigenvalue stays at 0.8.

spectra · Non-normality

Named alongside it

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

Complex conjugate pairCondition numberEigenvalue condition numberNon-normal matricesNon-normalityOrthogonal similarityOrthogonalityPerturbationPseudospectrumQuasi-triangularResolventShifts

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