Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Eigenvalue condition number — where it appears

The reciprocal of the inner product of an eigenvalue's left and right eigenvectors, which is one for a normal matrix and unbounded for a defective one.

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

110¹10²10³110¹10²10³off-diagonal entry ccondition number of the eigenvalue√(1 + c²)decoupled: 1measuredthree routes, one number‖A − ZTZᵀ‖/‖A‖1.7·10⁻¹⁵closed form100computed 1/|yᵀx|100worst measured movement46four eigenvalues, two conditioning numbersthe symmetric case has one, and it is 1

A condition number for one eigenvalue

In the symmetric case every eigenvalue has condition number exactly one. In this four-by-four matrix two of them have condition number 100.005 and the other two have exactly 1, and the number belongs to the eigenvalue rather than to the matrix.

spectra · eigen conditioning 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.

Non-normalityPerturbationCondition numberDeparture from normalityLeft eigenvectorNon-normal matricesOrthogonalityPseudospectrumThe real Schur formResolventSensitivitySingular values

All concepts