Matrix Analysis

Horn, Roger A.; Johnson, Charles R.

doi:10.1017/cbo9781139020411

Cited by 3,673 publications

(821 citation statements)

References 0 publications

Supporting

Mentioning

802

Contrasting

Unclassified

Order By: Relevance

“…As is seen from Eqs. (8) and (9), traces of positive-definite matrices [37] of the type AA ? stand in these relations.…”

Section: Theorymentioning

confidence: 99%

A simple rationale for lowered stabilities of branched and cross-conjugated polyenes

Gineitytė

2016

Monatsh Chem

View full text Add to dashboard Cite

The study is aimed at revealing the decisive factors for relative stabilities of acyclic p-electron systems of polyenes, the carbon backbones of which are of different type of branching. The systems are modeled as sets of N weakly interacting double (C=C) bonds. The relevant total p-electron energies are represented in the form of power series containing members of even orders with respect to the averaged resonance parameter of single (C-C) bonds. For distinct isomers of the same polyene, both zero-order energies and respective second-order corrections are shown to take uniform values. Relative stabilities of these isomers are then primarily determined by the fourth-order member of the series that, in turn, consists of two additive components of opposite signs, viz., of the stabilizing component expressible as a sum of transferable increments of individual triplets of linearly conjugated C=C bonds [i.e., of the three-membered conjugated paths (CPs)] and of the destabilizing component depending on overall adjacencies (connectivity) of C=C bonds. Lower stabilities of p-electron systems of branched and/or crossconjugated polyenes vs. the linear ones then follow from comparative analyses of the relevant fourth-order energies, and this destabilization is shown to originate either from (a) a reduced number of CPs or from (b) higher adjacencies of C=C bonds in the former isomers. An actual interplay of both factors (c) also is rather common. The three cases (a)-(c) are illustrated by specific examples. Graphical abstract

show abstract

“…As is seen from Eqs. (8) and (9), traces of positive-definite matrices [37] of the type AA ? stand in these relations.…”

Section: Theorymentioning

confidence: 99%

A simple rationale for lowered stabilities of branched and cross-conjugated polyenes

Gineitytė

2016

Monatsh Chem

View full text Add to dashboard Cite

show abstract

“…For nonnegative matrices, we have the following properties (see [1], pp.487-528, and [7], pp.661-687):…”

Section: Properties Of Nonnegative Matricesmentioning

confidence: 99%

“…A square matrix A is said to be idempotent [1] if and only if A 2 = A. The concept of r-potent matrices [2] is a generalization of idempotent matrices where a matrix A is said to be r-potent, for some natural number r, if and only if A r = A.…”

Section: Introductionmentioning

confidence: 99%

DECOMPOSABILITY OF NONNEGATIVE r-POTENT MATRICES

Thukral¹,

Marwaha²

2014

Int. J. of Pure and Appl. Math.

View full text Add to dashboard Cite

show abstract

“…Here, we have used the properties of positive definite and partitioned matrices (Horn and Johnson 1990), the fact that…”

Section: Property 5 the Block Coherence Reduces To Multiple Coherencementioning

confidence: 99%

Block coherence: a method for measuring the interdependence between two blocks of neurobiological time series

2011

View full text Add to dashboard Cite

Multisensor recordings are becoming commonplace. When studying functional connectivity between different brain areas using such recordings, one defines regions of interest, and each region of interest is often characterized by a set (block) of time series. Presently, for two such regions, the interdependence is typically computed by estimating the ordinary coherence for each pair of individual time series and then summing or averaging the results over all such pairs of channels (one from block 1 and other from block 2). The aim of this paper is to generalize the concept of coherence so that it can be computed for two blocks of non-overlapping time series. This quantity, called block coherence, is first shown mathematically to have properties similar to that of ordinary coherence, and then applied to analyze local field potential recordings from a monkey performing a visuomotor task. It is found that an increase in block coherence between the channels from V4 region and the channels from prefrontal region in beta band leads to a decrease in response time.

show abstract

Matrix Analysis

Cited by 3,673 publications

References 0 publications

A simple rationale for lowered stabilities of branched and cross-conjugated polyenes

A simple rationale for lowered stabilities of branched and cross-conjugated polyenes

DECOMPOSABILITY OF NONNEGATIVE r-POTENT MATRICES

Block coherence: a method for measuring the interdependence between two blocks of neurobiological time series

Contact Info

Product

Resources

About