Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Sensitivity — where it appears

2 essays name this object, across 2 fields. What follows is each of them, and the objects they name alongside it.
110¹10²10³10⁴10⁵10⁶10⁷00.250.50.751amplification of the input perturbationfraction of directions at or belowκ = 10·10⁵worst found 7.6·10⁵6×6, 200 directionsmedian reaches 0.29 of κ

The condition number is an amplifier

κ is usually introduced as a definition and then quoted. It is a measurement: perturb the input by a known amount, look at how much the output moves, and the largest ratio you can find is the number.

error · conditioning 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

Named alongside it

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

Condition numberOrthogonalityPerturbationDeparture from normalityEigenvalue condition numberLeft eigenvectorNon-normalitySingular valuesWorst-case analysis

All concepts