Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Jordan form — where it appears

The form a square matrix reaches under similarity, with each eigenvalue carried by blocks that hold it on the diagonal and ones just above. A block larger than one marks a defective eigenvalue, whose computed value moves by a root of a perturbation rather than in proportion to it.

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

-1-0.500.511.50eigenvalue of P⁻¹Ktriangular [[H, Aᵀ], [0, −Ŝ]] — 1 valuediagonal blkdiag(H, Ŝ) — 3 valuessteps to a residual of 10⁻¹⁰GMRES, triangular2MINRES, diagonal3‖P⁻¹K − I‖54computed |λ − 1| at c = 18.1·10⁻⁸one copy of each value against twoand the counts follow

One eigenvalue and two steps

Put the off-diagonal block back into a block-diagonal saddle-point preconditioner and every eigenvalue of the preconditioned matrix becomes exactly one. GMRES still needs two steps, because the matrix is the identity plus a nilpotent part of norm 54, and a computed eigenvalue at one comes back as a ring of radius 8·10⁻⁸ — the square root of the rounding, not the rounding. With an approximate Schur complement the triangular form leaves one copy of each value where the diagonal form leaves two, and the step count halves.

constraint · Block preconditioning 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in eᴬan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A_δ − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop

A function of a matrix is not a function of its entries

Everybody learns that f(A) means diagonalise, apply f to the eigenvalues, undiagonalise. That is a definition, not a method. On a matrix seven picometres from a defective one — with exact eigenvalues and eigenvectors from a closed form — the definition returns an answer wrong by sixty-five orders of magnitude, and a method that never mentions an eigenvalue returns the right one.

spectra · Matrix function

Named alongside it

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

Defective matrixBlock preconditionerCondition numberEigenvaluesEigenvectorsExact ground truthGMRESKrylov subspaceMatrix exponentialMatrix functionMinimal polynomialMINRES

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