Where does pivot collide?

In numerical linear algebra, the pivot is the element used to clear the entries below or beside it while solving a linear system by Gaussian elimination, and its size controls how rounding errors grow. Von Neumann and Goldstine's 1947 Numerical inverting of matrices of high order analysed the elimination process for solving and inverting large systems and the effect of the leading element on accumulated error, and Wilkinson's 1961 Error analysis of direct methods of matrix inversion established partial and complete pivoting, the rule of swapping rows or columns so the largest available element becomes the pivot. Choosing a well-sized pivot keeps the computed answer close to the true one. The referent is a chosen number inside a matrix computation, not a point a body turns on, a shift in strategy, or a decisive voter.[1][2]

Unit

One matrix element selected at each elimination step

Recovery means

Not applicable

Measured by

Judged by its magnitude relative to the other entries in its column or submatrix, which bounds the growth of rounding error

In political science and social choice, a pivotal actor, or pivot, is the voter or member whose position decides whether a proposal wins. Shapley and Shubik's 1954 A method for evaluating the distribution of power in a committee system defined a member as pivotal when joining an ordered coalition turns it from losing to winning, and measured each member's power by how often they hold that position. Krehbiel's 1998 Pivotal politics carried the idea into legislatures, identifying the filibuster and veto pivots whose positions fix whether a law can change. The referent is the decisive participant in a collective decision, not a point of rotation, an element in a matrix, or a firm's change of plan.[3][4]

Unit

One actor identified as decisive within a voting body

Recovery means

Not applicable

Measured by

Identified from members' positions and the decision rule, and summarised by power indices such as how often an actor is pivotal across orderings

In entrepreneurship, a pivot is a deliberate, structured change of a venture's strategy while keeping its underlying vision. Eric Ries's 2011 The lean startup named the pivot as a course correction made to test a new fundamental hypothesis about a product, business model, or engine of growth after early evidence falls short. Kirtley and O'Mahony's 2023 study in the Strategic Management Journal, What is a pivot?, examined when and how entrepreneurial firms actually decide to make such a change. The referent is a reorientation of a firm's plan, not a fixed point of rotation, a matrix element, or a decisive voter.[5][6]

Unit

One strategic reorientation of a single venture

Recovery means

Not applicable

Measured by

Identified from changes in a venture's product, customer, or business model, and studied through case histories and founder accounts