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Concept

Quadratic form

A homogeneous quadratic expression in several variables, written as a symmetric matrix sandwiched between a vector and itself. Its level sets are the conics and quadrics, and the signs of its eigenvalues decide which - a classification unchanged by any invertible change of coordinates.

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

The level curve, and the axes the matrix chooses. The curve xᵀAx = 1 for the matrix [2, 0.8, 0.8, 1.4], drawn by solving for the radius at each angle, with the two eigen-directions marked; they cross at a right angle and are the axes of the curve.

Symmetry forces a right angle

A matrix equal to its own reflection across the diagonal always has real stretches and always has perpendicular directions to stretch along. Neither is true of matrices in general, and both follow from one line of algebra.

algebra · Eigenvectors One sign decides which curve it is. 3 conics drawn from the general quadratic, each labelled with its discriminant B² − 4AC and the curve that sign names, checked against how many times the curve meets a large circle.

One sign decides which curve

The general quadratic in two variables has six coefficients and draws a conic. Which of the four it draws is settled by a single combination of three of them, and the other three cannot change the answer however they are chosen.

geometry · Conic sections

Named alongside it

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

EigenvalueBasisConicConic sectionDegenerate conicDiagonalisationDiscriminantEigenvectorEllipseHyperbolaInner productOrthogonality

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