Core of a matrix in max algebra and in nonnegative algebra: A survey

2019 ◽  
pp. 252-271
Author(s):  
Peter BUTKOVIC ◽  
Hans SCHNEIDER ◽  
Sergei SERGEEV
Keyword(s):  
Linear Algebra ◽  
Ergodic Theory ◽  
Nonnegative Matrix ◽  
Max Algebra ◽  
Similarities And Differences

This paper presents a light introduction to Perron-Frobenius theory in max algebra and in nonnegative linear algebra, and a survey of results on two cores of a nonnegative matrix. The (usual) core of a nonnegative matrix is defined as ∩ k≥1 span+ (A k ) , that is, intersection of the nonnegative column spans of matrix powers. This object is of importance in the (usual) Perron-Frobenius theory, and it has some applications in ergodic theory. We develop the direct max-algebraic analogue and follow the similarities and differences of both theories.

scholarly journals TEACHING LINEAR ALGEBRA TO STUDENTS OF ECONOMIC AND TECHNICAL SPECIALTIES: SIMILARITIES AND DIFFERENCES

2021 ◽  
Author(s):  
Evgenia V. VOLODİNA ◽  
Elena G. EFİMOVA ◽  
Alevtina G. KULAGİNA ◽  
Olympiada L. TARANOVA ◽  
Marina E. SİROTKİNA ◽  
...  
Keyword(s):  
Linear Algebra ◽  
Similarities And Differences

scholarly journals Z-matrix equations in max-algebra, nonnegative linear algebra and other semirings

2012 ◽  
Vol 60 (10) ◽  
pp. 1191-1210 ◽  
Author(s):  
Peter Butkovič ◽  
Hans Schneider ◽  
Sergeĭ Sergeev
Keyword(s):  
Linear Algebra ◽  
Matrix Equations ◽  
Max Algebra

scholarly journals TEACHING LINEAR ALGEBRA TO STUDENTS OF ECONOMIC AND TECHNICAL SPECIALTIES: SIMILARITIES AND DIFFERENCES

2021 ◽  
Author(s):  
Evgenia V. Volodina ◽  
Elena G. Efimova ◽  
Alevtina G. Kulagina ◽  
Olympiada L. Taranova ◽  
Marina E. Sirotkina ◽  
...  
Keyword(s):  
Linear Algebra ◽  
Similarities And Differences

scholarly journals SPN Graphs

2019 ◽  
Vol 35 ◽  
pp. 376-386
Author(s):  
Leslie Hogben ◽  
Naomi Shaked-Monderer
Keyword(s):  
Linear Algebra ◽  
Nonnegative Matrix ◽  
Positive Semidefinite ◽  
Simple Graph ◽  
Subdivision Graph ◽  
Copositive Matrix

A simple graph G is an SPN graph if every copositive matrix having graph G is the sum of a positive semidefinite and nonnegative matrix. SPN graphs were introduced in [N. Shaked-Monderer. SPN graphs: When copositive = SPN. Linear Algebra Appl., 509:82{113, 2016.], where it was conjectured that the complete subdivision graph of K4 is an SPN graph. This conjecture is disproved, which in conjunction with results in the Shaked-Monderer paper show that a subdivision of K_4 is a SPN graph if and only if at most one edge is subdivided. It is conjectured that a graph is an SPN graph if and only if it does not have an F_5 minor, where F_5 is the fan on five vertices. To establish that the complete subdivision graph of K_4 is not an SPN graph, rank-1 completions are introduced and graphs that are rank-1 completable are characterized.

scholarly journals Optimal Gersgorin-style estimation of the largest singular value. II

2016 ◽  
Vol 31 ◽  
pp. 679-685
Author(s):  
Charles Johnson ◽  
J. Pena ◽  
Tomasz Szulc
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Nonnegative Matrix ◽  
Singular Value ◽  
Nonnegative Matrices ◽  
Electronic Journal ◽  
Complex Matrix ◽  
Class Of Matrices

In estimating the largest singular value in the class of matrices equiradial with a given $n$-by-$n$ complex matrix $A$, it was proved that it is attained at one of $n(n-1)$ sparse nonnegative matrices (see C.R.~Johnson, J.M.~Pe{\~n}a and T.~Szulc, Optimal Gersgorin-style estimation of the largest singular value; {\em Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Next, some circumstances were identified under which the set of possible optimizers of the largest singular value can be further narrowed (see C.R.~Johnson, T.~Szulc and D.~Wojtera-Tyrakowska, Optimal Gersgorin-style estimation of the largest singular value, {\it Electronic Journal of Linear Algebra Algebra Appl.}, 25:48--59, 2011). Here the cardinality of the mentioned set for $n$-by-$n$ matrices is further reduced. It is shown that the largest singular value, in the class of matrices equiradial with a given $n$-by-$n$ complex matrix, is attained at one of $n(n-1)/2$ sparse nonnegative matrices. Finally, an inequality between the spectral radius of a $3$-by-$3$ nonnegative matrix $X$ and the spectral radius of a modification of $X$ is also proposed.

Factor rank preservers of matrices over additively-idempotent multiplicatively-cancellative semirings

2020 ◽  
pp. 2150072
Author(s):  
Sushobhan Maity ◽  
A. K. Bhuniya
Keyword(s):  
Boolean Algebra ◽  
Linear Algebra ◽  
Linear Operators ◽  
Multilinear Algebra ◽  
Boolean Rank ◽  
Max Algebra ◽  
Linear Multilinear Algebra

Here, we characterize the linear operators that preserve factor rank of matrices over additively-idempotent multiplicatively-cancellative semirings. The main results in this paper generalize the corresponding results on the two element Boolean algebra [L. B. Beasley and N. J. Pullman, Boolean-rank-preserving opeartors and Boolean-rank-1 spaces, Linear Algebra Appl. 59 (1984) 55–77] and on the max algebra [R. B. Bapat, S. Pati and S.-Z. Song, Rank preservers of matrices over max algebra, Linear Multilinear Algebra 48(2) (2000) 149–164]; and hold on max-plus algebra and some other tropical semirings.

Orosensory Perception, Speech Production, and Deafness

1973 ◽  
Vol 16 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Milo E. Bishop ◽  
Robert L. Ringel ◽  
Arthur S. House
Keyword(s):  
High School ◽  
Speech Production ◽  
Poor Performance ◽  
Normal Hearing ◽  
The Other ◽  
Functional Disorders ◽  
Form Discrimination ◽  
School Subjects ◽  
Oral Form ◽  
Similarities And Differences

The oral form-discrimination abilities of 18 orally educated and oriented deaf high school subjects were determined and compared to those of manually educated and oriented deaf subjects and normal-hearing subjects. The similarities and differences among the responses of the three groups were discussed and then compared to responses elicited from subjects with functional disorders of articulation. In general, the discrimination scores separated the manual deaf from the other two groups, particularly when differences in form shapes were involved in the test. The implications of the results for theories relating orosensory-discrimination abilities are discussed. It is postulated that, while a failure in oroperceptual functioning may lead to disorders of articulation, a failure to use the oral mechanism for speech activities, even in persons with normal orosensory capabilities, may result in poor performance on oroperceptual tasks.

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