On manifestations of the Schur complement

SuntoIn questo lavoro l’autore presenta alcuni modi in cui il complemento di Schur può essere usato in problemi numerici dell’algebra lineare.Si accenna dapprima all’eliminazione di variabili ed al «block pivoting»: la proprietà quoziente è successivamente usata per controllare il segno dei minori principali di una matrice. Un'altra interessante applicazione è il calcolo dell’inerzia di una matrice simmetrica reale: l’inerzia è utile per controllare in queste matrici la positiva (semi-) definitività.Questo risultato può essere sfruttato in problemi di programmazione matematica per controllare se una funzione non convessa è quasi convessa (pseudo convessa) nell’ortante non negativo.SummaryIn this paper the author is concerned with some of the ways in which the Schur complement can be used in numerical linear algebra.Variable elimination and block pivoting are first outlined: the quotient property is next used in order to test whether the leading principal minors of a matrix are nonzero or of a particular sign. Another useful application is in computing the inertia of a real symmetric matrix: the inertia is instrumental in checking such matrices for positive (semi-) definiteness.This can be exploited in mathematical programming problems in order to check a non convex quadratic function for quasi-convexity (pseudo-convexity) on the nonnegative orthant.

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