Concept

Positive definite

The requirement that a rule for multiplying two vectors give a strictly positive answer whenever a vector is multiplied by itself, unless the vector is zero. It is what turns such a rule into a geometry, since it makes the self-product a squared length, and it is the property whose loss reverses the Cauchy-Schwarz inequality.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The parabola that proves |a·b| ≤ |a||b|. The squared length of a − t b plotted against t. It is a parabola opening upward whose least value is 7.118; that this is never negative is exactly the Cauchy–Schwarz inequality.

The square that cannot be negative

Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.

algebra · Inner product

One subtraction clears a direction

A basis is a set of directions to measure along, and most bases are awkward because the directions get in each other's way. Removing one shadow at a time turns any basis into one where every coordinate is a shadow and nothing interferes.

algebra · Inner product

The map moved to the other side of the product. Two panels over the unit curve of the product u₁v₁ + u₂v₂. The left applies A to u and measures it against v, giving 2.328; the right applies the adjoint to v and measures it against u, giving the same number.

Moving a map across a product

The transpose looks like a fact about a matrix: reflect its entries in the diagonal. It is a fact about the inner product. Measure lengths and angles differently and the map that slides to the other side of the product is a different matrix, and a matrix that was symmetric stops being so.

algebra · Inner product

Every finite kaleidoscope on up to eight mirrors. 1: A₁; 2: A₂ B₂ H₂ G₂; 3: A₃ B₃ H₃; 4: D₄ A₄ B₄ H₄ F₄; 5: D₅ A₅ B₅; 6: E₆ D₆ A₆ B₆; 7: E₇ D₇ A₇ B₇; 8: E₈ D₈ A₈ B₈.

One determinant for every kaleidoscope

Three mirrors close up into a kaleidoscope when the angles of their triangle add to more than half a turn. In more dimensions there is no triangle to add up, but there is a matrix — the inner products of the mirrors' directions — and the mirrors close up exactly when it is positive definite. Trying every arrangement of up to nine mirrors against that one test finds Coxeter's list: three families in every dimension, a fork that runs out at eight mirrors because its determinant reaches zero at nine, and five exceptions. The same search finds the hyperbolic kaleidoscopes, and finds that they stop at five mirrors.

geometry · Regular polyhedra

Named alongside it

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

Inner product Orthogonality Cauchy schwarz Dot product Least squares Projection Residual Subspace Adjoint Approximation Basis Coxeter diagram

All concepts