Concept

Spectral theorem

The theorem that a map equal to its own adjoint has real eigenvalues and eigenvectors perpendicular to one another. It holds under whichever inner product the map is self-adjoint for, and it is why symmetric matrices, reversible chains and Fourier series all split into independent directions.

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.

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

The map moved to the other side of the product. Two panels over the unit curve of the product u₁v₁ + u₂v₂. The left applies A to u and measures it against v, giving 2.328; the right applies the adjoint to v and measures it against u, giving the same number.

Moving a map across a product

The transpose looks like a fact about a matrix: reflect its entries in the diagonal. It is a fact about the inner product. Measure lengths and angles differently and the map that slides to the other side of the product is a different matrix, and a matrix that was symmetric stops being so.

algebra · Inner product

Named alongside it

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

Eigenvector Inner product Orthogonality Adjoint Basis Conic section Diagonalisation Eigenvalue Gradient Least squares Markov chain Positive definite

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