Existence of Two-Point Oscillatory Solutions of a Relay Nonautonomous System with Multiple Eigenvalue of a Real Symmetric Matrix

2021 ◽  
Author(s):  
V. V. Yevstafyeva
Keyword(s):  
Symmetric Matrix ◽  
Nonautonomous System ◽  
Multiple Eigenvalue ◽  
Oscillatory Solutions ◽  
Real Symmetric Matrix

scholarly journals Cauchy's interlace theorem and lower bounds for the spectral radius

2000 ◽  
Vol 23 (8) ◽  
pp. 563-566 ◽  
Author(s):  
A. McD. Mercer ◽  
Peter R. Mercer
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Simple Proof ◽  
Symmetric Matrix ◽  
Real Symmetric Matrix

We present a short and simple proof of the well-known Cauchy interlace theorem. We use the theorem to improve some lower bound estimates for the spectral radius of a real symmetric matrix.

scholarly journals A shorter proof of the distance energy of complete multipartite graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 61-63 ◽  
Author(s):  
Wasin So
Keyword(s):  
Linear Algebra ◽  
Symmetric Matrix ◽  
Complete Multipartite Graph ◽  
Real Symmetric Matrix ◽  
Multipartite Graphs ◽  
Complete Multipartite Graphs ◽  
Multipartite Graph ◽  
Complete Multipartite

Abstract Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009-64] conjectured that the distance energy of a complete multipartite graph of order n with r ≥ 2 parts, each of size at least 2, is equal to 4(n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities of a positve semi-definite rank-1 perturbation to a real symmetric matrix.

scholarly journals A modified simplex partition algorithm to test copositivity

2021 ◽  
Author(s):  
Mohammadreza Safi ◽  
Seyed Saeed Nabavi ◽  
Richard J. Caron
Keyword(s):  
Symmetric Matrix ◽  
Maximum Clique ◽  
Numerical Results ◽  
Maximum Clique Problem ◽  
Real Symmetric Matrix ◽  
Standard Simplex ◽  
The Creation ◽  
Clique Problem

AbstractA real symmetric matrix A is copositive if $$x^\top Ax\ge 0$$ x ⊤ A x ≥ 0 for all $$x\ge 0$$ x ≥ 0 . As A is copositive if and only if it is copositive on the standard simplex, algorithms to determine copositivity, such as those in Sponsel et al. (J Glob Optim 52:537–551, 2012) and Tanaka and Yoshise (Pac J Optim 11:101–120, 2015), are based upon the creation of increasingly fine simplicial partitions of simplices, testing for copositivity on each. We present a variant that decomposes a simplex $$\bigtriangleup $$ △ , say with n vertices, into a simplex $$\bigtriangleup _1$$ △ 1 and a polyhedron $$\varOmega _1$$ Ω 1 ; and then partitions $$\varOmega _1$$ Ω 1 into a set of at most $$(n-1)$$ ( n - 1 ) simplices. We show that if A is copositive on $$\varOmega _1$$ Ω 1 then A is copositive on $$\bigtriangleup _1$$ △ 1 , allowing us to remove $$\bigtriangleup _1$$ △ 1 from further consideration. Numerical results from examples that arise from the maximum clique problem show a significant reduction in the time needed to establish copositivity of matrices.

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