zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Reduction to tridiagonal form and minimal realizations. (English) Zbl 0754.65040

The paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix.

In the preparative part of the paper the author describes the representation of the class of similar tridiagonals by vector pairs and the use of a pair (T ^,Ω), with T ^ symmetric tridiagonal and Ω diagonal, rather than a single matrix Ω -1 T.

The fundamental result of the paper says that the tridiagonal reduction is equivalent to a Gram-Schmidt process applied to two Krylow sequences. In Euclidean space a proper normalization allows one to monitor a tight lower bound on the condition number of the transformation.

MSC:
65F30Other matrix algorithms
65F15Eigenvalues, eigenvectors (numerical linear algebra)
65K10Optimization techniques (numerical methods)
15A21Canonical forms, reductions, classification
93B10Canonical structure of systems