Every essay — page 9
The matrix a constraint makes
Every difficult matrix in the other fields was difficult for a reason the arithmetic supplied: a Hilbert matrix arrives ill conditioned, a Wilkinson matrix grows under elimination, a kernel matrix is dense. Ask a problem to minimise something subject to a constraint and the matrix that results is difficult for a reason the algebra supplies. Its zero block is the second derivative of a Lagrangian with respect to its own multipliers, so no pivot order removes it and no precision changes that; its inertia is known before anything runs; and Cholesky does not fail somewhere on it, it fails at the first constraint row, on a number the problem already contained. Then the field's second surprise, which is the opposite one: an interior-point method drives the condition number of this matrix to 10¹⁵ deliberately, and the answer keeps fifteen digits — because the number that describes the error is not the one every library prints.
A condition number sent to infinity
An interior-point method manufactures an ill-conditioned matrix on every iteration, deliberately, because the separating of a diagonal is how it discovers which constraints are active. Written one way the answer keeps fifteen digits at a condition number of 3·10¹⁵. Written the other way — the way almost every code writes it — it has none left.
The regularisation that legalises every order
Perturb a saddle-point matrix's two blocks in opposite directions and it acquires a factorisation with a diagonal D under every symmetric permutation — not under a good one, under all of them. Five hundred random orderings, five hundred successes, and a growth factor that spans six orders across them.