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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