Symmetric matrix
Named by 6 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.](/f/the-level-curve-and-the-axes-the-matrix-chooses-81b169af.svg)
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.
![What the map does to a circle. The unit circle with two perpendicular directions marked, and its image under [1.6, 1.2, −0.4, 1.1] — an ellipse whose axes are the images of those two directions, of lengths 2.04 and 1.10.](/f/what-the-map-does-to-a-circle-29ac1dac.svg)
What a map does to a circle
Every linear map sends the unit circle to an ellipse. Two perpendicular directions go to two perpendicular directions, whatever the map is — even a map with no invariant direction at all, and even one that is not square.
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.
Every light on, from every light off
In Lights Out, pressing a light toggles it and its neighbours. On a five-by-five board only one pattern in four can be reached from all off — and yet all on is always one of them. The same is true on every board and on every network of lights whatever, provided the influence runs both ways. The proof is a single observation about symmetric matrices over a field with two elements, and it fails, as a three-arrow example shows, the moment influence runs one way.
When a press skips its own light
In Lights Out every board can be turned from all off to all on, because each press toggles its own light. Change the rule so a press toggles only the neighbours and the guarantee is gone: 310 of the 3,600 boards up to sixty by sixty cannot be lit. A single comparison of powers of two predicts every one of them, and on general graphs the answer depends on whether the number of vertices is odd or even, for a reason that is one line of algebra.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
EigenvalueEigenvectorFinite fieldInterlacingKernelLinear systemOrthogonalityParityQuadratic formRankSpectral theoremApproximation