The Schur Complement and Its ApplicationsFuzhen Zhang Springer Science & Business Media, 30 de març 2006 - 295 pàgines What's in a name? To paraphrase Shakespeare's Juliet, that which - ilie Haynsworth called the Schur complement, by any other name would be just as beautiful. Nevertheless, her 1968 naming decision in honor of Issai Schur (1875-1941) has gained lasting acceptance by the mathematical com munity. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications. Our goal is to expose the Schur complement as a rich and basic tool in mathematical research and applications and to discuss many significant re sults that illustrate its power and fertility. Although our book was originally conceived as a research reference, it will also be useful for graduate and up per division undergraduate courses in mathematics, applied mathematics, and statistics. The contributing authors have developed an exposition that makes the material accessible to readers with a sound foundation in linear algebra. The eight chapters of the book (Chapters 0-7) cover themes and varia tions on the Schur complement, including its historical development, basic properties, eigenvalue and singular value inequalities, matrix inequalities in both finite and infinite dimensional settings, closure properties, and appli cations in statistics, probability, and numerical analysis. The chapters need not be read in the order presented, and the reader should feel at leisure to browse freely through topics of interest. |
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A₁ Aitken algorithm Applications B₁ Banachiewicz block matrix Brezinski Broyden C-matrix Closure Cmxn Cnxn coefficients compute condition consider Corollary correlation matrix covariance matrix Cramér-Rao Inequality decomposition defined definite matrix denote eigenvalues Emilie entries equal equations equivalent full column rank Hadamard product Haynsworth hence hereditary Hermitian matrix Hilbert space identity In(A index set inertia inverse class inverse-closed inversion formula Issai Schur Lemma Linear Algebra linear model Löwner partial ordering M₁ Mathematics matrix inequalities methods n x n nonnegative definite nonsingular nonzero notation obtain orthogonal polynomials orthoprojection P₁ Padé approximants partitioned positive definite positive semidefinite matrices principal submatrix Proof quotient formula quotient property ran(A recursive SC-closed SC-closure scalar Schur complement Section sequence transformations similarity invariant singular values square statistics Styan subspace symmetric Theorem 5.3 triangular unitarily invariant norm unitary X₁ Xẞ