the min-max characterization of eigenvalues
The Rayleigh quotient gives the lowest eigenvalue as a minimum. But how do you reach the second, third, and higher eigenvalues by minimizing, without already knowing the earlier eigenfunctions? The min-max characterization answers this: it pins down every lambda_n purely as the outcome of a minimization-and-maximization game over subspaces, no eigenfunctions required in advance.
One clean version (the Courant-Fischer principle) says: lambda_n equals the minimum, over all n-dimensional subspaces V of admissible functions, of the maximum of the Rayleigh quotient R[u] over nonzero u in V. In words — you pick the best n-dimensional 'spread' of trial functions, look at the worst (largest) Rayleigh value inside it, and the smallest such worst-case over all choices is exactly the n-th eigenvalue. A simpler related form: lambda_n is the minimum of R[u] over all u orthogonal to the first n-1 eigenfunctions.
Why this is powerful: it makes eigenvalues monotone and comparable without ever solving the equation. Shrink the domain or stiffen the operator and every R[u] can only go up, so every lambda_n goes up — a fact you can read straight off the formula. This is the engine behind eigenvalue comparison theorems, the Weyl asymptotic law for how eigenvalues grow, and rigorous numerical bounds. It is the same Courant-Fischer min-max that governs symmetric matrices, lifted to differential operators.
Domain monotonicity: enlarge a vibrating membrane and every natural frequency drops. Min-max explains it instantly — admissible functions on the small domain extend (by zero) to the big one, so the minimizing subspaces for the big domain include more candidates, the min-max value can only decrease, hence each lambda_n decreases. A bigger drum sounds lower, with no computation.
Min-max turns 'a bigger drum sounds lower' into a one-line rigorous proof.
There are two equivalent readings — a min-over-max (Courant-Fischer) and a max-over-min — and which is which depends on whether you index eigenvalues from the bottom up. Both give the same numbers; the key takeaway is that eigenvalues are extremal values, hence monotone under domain or coefficient changes.