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Concept

Interlacing

The pattern in which the roots of two real-rooted polynomials alternate along the line, one root of each between consecutive roots of the other. It is what deleting a vertex does to a matching polynomial and what removing a row and column does to a symmetric matrix's eigenvalues.

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

The largest value on each plane, over every direction of space. A longitude–latitude map of directions u in three dimensions, each shaded by the largest value of xᵀAx on the plane perpendicular to u. The smallest such value is the middle eigenvalue 1.829, reached at ±v₁; the largest is 3.622, reached along a great circle.

The highest point on the sphere is an eigenvalue

For a symmetric matrix, walk a unit arrow over every direction and record the value of xᵀAx. The highest value reached is the largest eigenvalue, the lowest is the smallest, and every eigenvalue in between is a saddle height, a minimum of maxima. From that one description comes a theorem no formula for the roots could give: delete a row and its column, and every eigenvalue of what is left sits between two of the original's.

algebra · Eigenvectors A spider's matchings have real roots and its independent sets do not. Spider with legs 3, 3, 3: matching polynomial 1, 9, 27, 32, 12, roots -0.5000, -0.2324, -0.5000, -1.4343; independence polynomial 1, 10, 36, 57, 38, 9, 1, 4 non-real roots.

The roots every matching polynomial keeps real

Count a graph's matchings by size and make the counts the coefficients of a polynomial. On every one of 35,664 graphs tested, that polynomial has only real roots — a theorem of Heilmann and Lieb from 1972. Count independent sets instead and the roots wander off the axis on a growing share of graphs, but never on a graph without a claw, and matchings are the independent sets of a graph that never has one.

discrete · Generating functions

Named alongside it

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

EigenvalueEigenvectorGenerating functionIndependent setLog-concavityOptimisationPerfect matchingQuadratic formRayleigh quotientReal rooted polynomialSpectral theoremSphere

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