Cauchy interlacing theorem
The Cauchy interlacing theorem describes what happens to the eigenvalues of a symmetric matrix when you delete a row and the matching column. The smaller matrix has fewer eigenvalues, but they cannot land anywhere — they must weave neatly between the eigenvalues of the full matrix, never straying outside the gaps.
Let A be a symmetric n-by-n matrix with eigenvalues lambda_1 >= ... >= lambda_n, and let B be the (n-1)-by-(n-1) principal submatrix obtained by deleting one row and the same column, with eigenvalues mu_1 >= ... >= mu_{n-1}. Interlacing says lambda_k >= mu_k >= lambda_{k+1} for every k. Each submatrix eigenvalue is sandwiched between two consecutive eigenvalues of the parent.
More generally, if B is an m-by-m principal submatrix of an n-by-n symmetric A, then lambda_{k+n-m} <= mu_k <= lambda_k. The proof again leans on Courant-Fischer: restricting the Rayleigh quotient to a subspace can only squeeze its max-min values inward, never expand them.
The payoff is both theoretical and computational. Interlacing tells you a leading submatrix can never have an eigenvalue larger than the whole matrix, it gives Sturm-sequence methods for counting eigenvalues in an interval, and it underlies why deflation and bisection eigensolvers are well behaved on symmetric problems.
The two eigenvalues of a 2x2 principal submatrix nest perfectly between the three of the parent 3x3.
Interlacing implies the largest eigenvalue only grows (and the smallest only shrinks) as you append rows and columns — a monotone bracketing that adaptive eigenvalue estimators exploit.