Symmetric matrix
Named by 2 essays across one field — 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-db3c3a9d.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-cd467b80.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.
Named alongside it
The objects these essays reach for when they reach for this one.
EigenvectorOrthogonalityApproximationBasisCondition numberConic sectionDeterminantDiagonalisationEigenvalueInner productMatrixQuadratic form