The numerical range of self-adjoint quadratic matrix polynomials

Lancaster, P; Psarrakos, P

dc.contributor.author	Lancaster, P	en
dc.contributor.author	Psarrakos, P	en
dc.date.accessioned	2014-03-01T01:18:25Z
dc.date.available	2014-03-01T01:18:25Z
dc.date.issued	2002	en
dc.identifier.issn	0895-4798	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/14999
dc.subject	matrix polynomial	en
dc.subject	numerical range	en
dc.subject	eigenvalue	en
dc.subject	boundary	en
dc.subject.classification	Mathematics, Applied	en
dc.title	The numerical range of self-adjoint quadratic matrix polynomials	en
heal.type	journalArticle	en
heal.identifier.primary	10.1137/S0895479899364088	en
heal.identifier.secondary	http://dx.doi.org/10.1137/S0895479899364088	en
heal.language	English	en
heal.publicationDate	2002	en
heal.abstract	Matrix polynomials of the form P (lambda) = Ilambda(2) + A(1)lambda + A(0) (where A(0) and A(1) are n x n Hermitian matrices and lambda is a complex variable) arise in many applications. The numerical range of such a polynomial is W(P) = {lambda epsilon C : x*P(lambda)x = 0 for some nonzero x epsilon C-n} and it always contains the spectrum of P(lambda), i. e., the set of zeros of detP(lambda). Properties of the numerical range are developed in detail, taking advantage of the close connection between W ( P) and the classical numerical range (field of values) of the ( general) complex matrix A : = A(0) + iA(1). Eigenvalues and nondifferentiable points on the boundary are examined and a procedure for the numerical determination of W(P) is presented and used for several illustrations. Some extensions of the theory to more general polynomials P(lambda) are also discussed, as well as special cases describing vibrating systems.	en
heal.publisher	SIAM PUBLICATIONS	en
heal.journalName	SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS	en
dc.identifier.doi	10.1137/S0895479899364088	en
dc.identifier.isi	ISI:000174378500002	en
dc.identifier.volume	23	en
dc.identifier.issue	3	en
dc.identifier.spage	615	en
dc.identifier.epage	631	en