Eigendecomposition of a real symmetric matrix

Every real symmetric matrix can be decomposed using only real eigenvectors and eigenvalues: A = QΛQᵀ. Q is an orthogonal matrix whose columns are eigenvectors of A, and Λ is a diagonal matrix holding the matching eigenvalues. Geometrically, A scales space by λᵢ in the direction of the i-th eigenvector.

By convention the eigenvalues in Λ are sorted in descending order; the decomposition is then unique only if all the eigenvalues are distinct.

It also answers an optimization question. Over unit vectors (‖x‖₂ = 1), f(x) = xᵀAx is largest at the eigenvector with the largest eigenvalue, where it equals that eigenvalue, and smallest at the eigenvector with the smallest one.