Concept

Null-space basis — where it appears

A matrix Z whose columns span every vector a constraint matrix A sends to zero, so each solution of Ax = b is one particular solution plus Z times some vector. Every such Z describes the same space; they differ in conditioning, sparsity and the cost of building them.

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

The basis nobody chose on purpose

A method that eliminates a constraint has to pick a basis for its null space, and every basis is correct. Their condition numbers are eight orders apart, the reduced problem inherits the square, and the choice is usually made by a one-line rule nobody thought of as a numerical decision.

orthogonality · Null-space basis

The tree the resistances choose

On a network every basic set is a spanning tree and every null-space basis is a set of loops with entries 0 and ±1, so no tree can make Z badly conditioned. The tree with the best-conditioned Z still gives loop equations 4.8 times worse than the tree of least resistance: the basis has to be chosen against the Hessian, and pivoting finds it only when it pivots on the resistances too.

orthogonality · Null-space basis

Named alongside it

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

Column pivoting Condition number Null-space Reduced hessian Basis Constrained minimisation Graph laplacian Grounding Null-space method Orthonormal basis QR factorisation Saddle-point systems

All concepts