8 1. SOME FUNDAMENTALS S1. All the eigenvalues of A are real numbers, and thus can be ordered as λmin = λn ≤ λn−1 ≤ . . . ≤ λ1 = λmax. Also there exist eigenvectors x1,x2,...,xn of A for λ1,λ2,...,λn, respectively, that form an orthonormal basis of n . S2. For every nonzero vector x ∈ n , λn ≤ xtAx xtx ≤ λ1. Equivalently, for every vector x ∈ n of length 1, that is, xtx = 1, λn ≤ xtAx ≤ λ1. In fact, λ1 = max{xtAx : xtx = 1} and λn = min{xtAx : xtx = 1}. S3. More generally, each of the eigenvalues of A can be characterized using subspaces of n . Let x1,x2,...,xn be an orthonormal basis of n consisting of eigenvectors of A for λ1,λ2,...,λn, respectively. Then for a nonzero vector y ∈ span{xi,...,xn} λi ≥ ytAy yty with equality for y = xi, and for y ∈ span{x1,...,xi}, λi ≤ ytAy yty with equality for y = xi. Thus λi = inf{sup ytAy yty : y ∈ W : W a subspace of n of dimension n − i + 1}. S4. (Interlacing of eigenvalues I.) Let Q be an n × m real matrix with orthonormal columns (QtQ = Im), and let the eigenvalues of the m × m matrix QtAQ be μ1 ≥ μ2 ≥ · · · ≥ μm. Then λn−m+i ≤ μi ≤ λi (i = 1, 2,...,m). S5. (Interlacing of eigenvalues II.) If μ1 ≥ μ2 ≥ · · · ≥ μm are the eigenvalues of an m × m principal submatrix of A, then λn−m+i ≤ μi ≤ λi (i = 1, 2,...,m). In particular, if μ1 ≥ μ2 ≥ · · · ≥ μn−1 are the eigenvalues of an n − 1 × n − 1 principal submatrix of A, then λn ≤ μn−1 ≤ λn−1 ≤ · · · ≤ μ2 ≤ λ2 ≤ μ1 ≤ λ1.

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