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)
Matrix diagonal stability and its implications. (English) Zbl 0547.15009

This paper considers inclusion relations between the following four classes of real square matrices:

The diagonally stable matrices 𝒜 consisting of those matrices A such that AD+DA T is positive definite for some positive definite diagonal matrix D, the positive stable matrices consisting of those matrices A such that AX+XA T is positive definite for some positive definite matrix X, the P-matrices 𝒫 consisting of those A all of whose principal minors are positive, and the semi-positive matrices 𝒮 consisting of all A such that Ax>0 for some x>0·

In particular, it is shown that if A is a P-matrix and if the nondirected graph of A is a forest, then A is diagonally stable. (The nondirected graph of an n×n matrix A is the graph whose vertices are 1,2,..., and whose edges are the pairs { i,j}, ij, for which either a ij or a ji is nonzero.) This result, in view of previously known results, yields for those matrices whose nondirected graph is a forest the relations 𝒜𝒫𝒮 where means ”is contained in” and the absence of an implication implies a counterexample. Partial results and some open questions are given for real spectra matrices and for a class of matrices called ω -matrices.

Reviewer: J.Brawley
MSC:
15A48Positive matrices and their generalizations (MSC2000)
15A18Eigenvalues, singular values, and eigenvectors
15A42Inequalities involving eigenvalues and eigenvectors