spectral theorem and positivity

Positivity for numbers means lying to the right of zero. The spectral theorem lets you lift that idea to operators in a way that behaves just as cleanly. A self-adjoint operator T is called positive (more carefully, positive semidefinite) when <Tv, v> >= 0 for every vector v. Geometrically T never points a vector back across itself: the output always has a nonnegative component along the input.

The spectral theorem turns this analytic condition into a transparent statement about the spectrum. A self-adjoint operator is positive if and only if all of its eigenvalues are nonnegative; it is strictly positive (positive definite) exactly when all eigenvalues are strictly greater than 0. The reason is immediate from the Rayleigh quotient: <Tv, v> / <v, v> ranges over [lambda_min, lambda_max], so the quadratic form stays nonnegative precisely when lambda_min >= 0. Positivity of an operator is positivity of its spectrum, nothing more.

This characterization is the gateway to a whole toolkit. Because every eigenvalue is real and nonnegative, the functional calculus can apply functions that only make sense on [0, infinity) — most importantly sqrt, giving a unique positive square root. It also explains why Gram matrices A^* A are always positive (their eigenvalues are squared singular values), why covariance matrices are positive, and why positivity is preserved under congruence. Positive operators are the operator analogue of nonnegative numbers, and the spectrum is what makes the analogy exact.