scholarly journals A Note on the Drazin Indices of Square Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-3
Author(s):  
Lijun Yu ◽  
Tianyi Bu ◽  
Jiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Drazin Index ◽  
Square Matrix

For a square matrixA, the smallest nonnegative integerksuch that rank (Ak) = rank (Ak+1) is called the Drazin index ofA. In this paper, we give some results on the Drazin indices of sum and product of square matrices.

Stochastic local operations and classical communication invariants via square matrix

Laser Physics ◽  
2019 ◽  
Vol 29 (2) ◽  
pp. 025203 ◽  
Author(s):  
Xinwei Zha ◽  
Irfan Ahmed ◽  
Da Zhang ◽  
Wen Feng ◽  
Yanpeng Zhang
Keyword(s):  
Classical Communication ◽  
Square Matrix

scholarly journals Algebraic conditions and the sparsity of spectrally arbitrary patterns

2021 ◽  
Vol 9 (1) ◽  
pp. 257-274
Author(s):  
Louis Deaett ◽  
Colin Garnett
Keyword(s):  
Major Goal ◽  
Algebraic Condition ◽  
Open Question ◽  
Algebraic Tool ◽  
General Method ◽  
Square Matrix ◽  
Spectrally Arbitrary

Abstract Given a square matrix A, replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern. Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A. A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2n. An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young
Keyword(s):  
Unit Circle ◽  
Matrix Polynomial ◽  
Hermitian Matrix ◽  
Krein Space ◽  
Gram Matrix ◽  
Natural Basis ◽  
Test Matrix ◽  
Matrix Polynomials ◽  
Rational Vector ◽  
Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

A trace bound for a general square matrix product

2000 ◽  
Vol 45 (8) ◽  
pp. 1563-1569 ◽  
Author(s):  
Wei Xing ◽  
Qingling Zhang ◽  
Qiyi Wang
Keyword(s):  
Matrix Product ◽  
Square Matrix

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

A Model of Secure Functioning of Computer Systems

2021 ◽  
Vol 12 (3) ◽  
pp. 150-156
Author(s):  
A. V. Galatenko ◽  
◽  
V. A. Kuzovikhina ◽  
Keyword(s):  
Lower Bound ◽  
Computer System ◽  
Finite Automaton ◽  
Nonnegative Integer ◽  
The Other ◽  
Shannon Function ◽  
Security Breaches ◽  
System Functioning ◽  
The Cost ◽  
The Given

We propose an automata model of computer system security. A system is represented by a finite automaton with states partitioned into two subsets: "secure" and "insecure". System functioning is secure if the number of consecutive insecure states is not greater than some nonnegative integer k. This definition allows one to formally reflect responsiveness to security breaches. The number of all input sequences that preserve security for the given value of k is referred to as a k-secure language. We prove that if a language is k-secure for some natural and automaton V, then it is also k-secure for any 0 < k < k and some automaton V = V (k). Reduction of the value of k is performed at the cost of amplification of the number of states. On the other hand, for any non-negative integer k there exists a k-secure language that is not k"-secure for any natural k" > k. The problem of reconstruction of a k-secure language using a conditional experiment is split into two subcases. If the cardinality of an input alphabet is bound by some constant, then the order of Shannon function of experiment complexity is the same for al k; otherwise there emerges a lower bound of the order nk.

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