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Concept

Semicircle law

Wigner's law that the eigenvalues of a large random symmetric matrix, suitably scaled, are spread with a semicircular density. It holds whatever the distribution of the entries, provided they are independent with equal variance.

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

Three ways to pick a rotation at random. Mean rotation angle: uniform 126.70° (exact 126.48°), uniform Euler angles 127.91°, uniform axis and angle 90.18°. Densities by 5° bin, uniform: 0.00, 0.00, 0.01, 0.01, 0.03, 0.04, 0.04, 0.06, 0.09, 0.09, 0.13, 0.15, 0.16, 0.20, 0.23, 0.25, 0.28, 0.29, 0.34, 0.36, 0.38, 0.41, 0.44, 0.46, 0.47, 0.52, 0.54, 0.55, 0.57, 0.61, 0.60, 0.64, 0.62, 0.61, 0.64, 0.64.

A rotation chosen at random

Turn a solid body to an orientation chosen at random and it has usually been turned a long way: on average 126.5 degrees, and by less than ten degrees only three times in ten thousand. Unit quaternions make the right meaning of 'at random' a picture, and two recipes that look fair fail it in opposite directions.

algebra · Quaternions Eigenvalues that will not crowd together. Unfolded spacings, eigenvalues: 0.36 1.52 1.15 0.88 1.23 1.13 0.34 0.75 2.05 0.65 1.66 0.64 1.07 1.01 0.86 1.23 0.69 1.30 0.77 0.94 0.99 0.71 0.92 2.49 0.32 1.07 0.62 1.34 0.07 0.51 0.98 2.25 0.64 0.78 0.35 0.95 1.36 1.89 0.21 0.79 0.46 0.84 1.12 1.52 0.96 0.65 1.09 0.29 1.30 1.06 1.83 1.50 0.28 0.43 1.19 1.01 0.52 1.54 1.51 1.56. Independent points: 0.34 0.01 0.31 3.72 0.25 2.29 0.98 2.14 1.99 0.54 0.15 0.58 0.96 0.22 0.26 1.11 0.04 0.38 0.55 0.74 0.89 0.11 1.47 0.52 0.06 2.11 1.29 3.04 0.44 0.28 2.49 3.97 0.87 1.04 0.83 0.77 0.79 0.69 1.76 1.82 0.14 0.07 1.03 0.88 1.05 0.37 0.06 0.82 0.70 1.72 0.90 1.24 0.32 0.25 1.07 0.55 1.59 3.23 0.36 0.40. Gaps under a quarter of the mean: 2 against 9.

Eigenvalues that keep their distance

Fill a large symmetric matrix with independent random numbers and its eigenvalues do something no list of independent numbers does: they refuse to crowd together. The reason is a count of conditions in a two-by-two matrix, and its consequences — a semicircle, gaps that vanish at nought, a spectrum almost as rigid as a lattice — reach from heavy nuclei to the zeros of the zeta function.

algebra · Eigenvectors

Named alongside it

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

Avoided crossingDouble coverEigenvalueIndependenceInterlacingInterpolationQuaternionRandom matrixRandom walkRotationSymmetric matrixSymmetry group

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