Positivity is a statement about the spectrum
A self-adjoint operator is positive semidefinite when <Ax, x> >= 0 for all x, and positive definite when the inequality is strict for x != 0. Via the spectral theorem this turns into a clean spectral statement: A is positive iff all its eigenvalues are nonnegative. The quadratic form <Ax, x> = sum lambda_i |<x, q_i>|^2 is a nonnegative combination of squares exactly when every lambda_i >= 0. This connects straight back to Vol I's positive definite matrix.
The unique positive square root
Positivity plus functional calculus gives a clean construction: the positive square root. Since every eigenvalue of a positive A is >= 0, the function sqrt is well-defined on the spectrum, so A^{1/2} = sum sqrt(lambda_i) P_i is itself a positive operator with (A^{1/2})^2 = A. And it is unique among positive operators — there is exactly one positive B with B^2 = A.
A = [2, 1; 1, 2] (positive: eigenvalues 3, 1, both > 0)
spectral resolution: A = 3 P_1 + 1 P_2
P_1 = [0.5, 0.5; 0.5, 0.5], P_2 = [0.5,-0.5;-0.5, 0.5]
positive square root applies sqrt eigenvalue-wise:
A^{1/2} = sqrt(3) P_1 + sqrt(1) P_2
= sqrt(3) [0.5,0.5;0.5,0.5] + [0.5,-0.5;-0.5,0.5]
verify: (A^{1/2})^2 = 3 P_1 + 1 P_2 = A (P_i orthogonal idempotents)
Why unique: any positive B with B^2 = A is diagonal in A's eigenbasis
and must take the NONnegative root on each eigenvalue -> forced to be A^{1/2}.The positive square root is not a curiosity — it is the engine behind the polar decomposition A = U |A| with |A| = (A* A)^{1/2}, and behind whitening transforms in statistics. Whenever you need a 'half' of a positive operator, the spectral theorem hands you exactly one.
Eigenvalues as optimization: Rayleigh and min-max
For a self-adjoint A the Rayleigh quotient R(x) = <Ax, x> / <x, x> is the average stretch of A in direction x. The spectral theorem pins its range exactly: R(x) ranges over [lambda_min, lambda_max], reaching the largest eigenvalue when x is the top eigenvector and the smallest when x is the bottom one. So the extreme eigenvalues are an optimization problem, not just roots of a polynomial.
The interior eigenvalues are captured too by the Courant-Fischer min-max theorem: the k-th eigenvalue is a min over k-dimensional subspaces of the max of the Rayleigh quotient on each (and dually, a max-min). This variational description is what makes eigenvalues *stable*: Weyl's inequalities say that perturbing A by E shifts each eigenvalue by at most ||E||, because each eigenvalue is an optimum of a quotient that itself moves only a little.