scholarly journals On idempotency of linear combinations of a quadratic or a cubic matrix and an arbitrary matrix

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3161-3185
Author(s):  
Murat Sarduvan ◽  
Nurgül Kalaycı
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Linear Combinations ◽  
Arbitrary Matrix ◽  
Necessary And Sufficient

Let A be a quadratic or a cubic n x n nonzero matrix and B be an arbitrary n x n nonzero matrix. In this study, we have established necessary and sufficient conditions for the idempotency of the linear combinations of the form aA + bB, under the some certain conditions imposed on A and B, where a, b are nonzero complex numbers.

On the Construction of Nonnegative 5×5 Matrices from Spectrum Data

2013 ◽  
Vol 444-445 ◽  
pp. 621-624
Author(s):  
Zhi Bing Liu ◽  
Zhen Tu ◽  
Cheng Feng Xu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Nonnegative Matrices ◽  
Necessary And Sufficient ◽  
Spectrum Data

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

scholarly journals Injectivity of linear combinations in $\mathcal{B}(\mathcal{H})$

2021 ◽  
Vol 37 ◽  
pp. 359-369
Author(s):  
Marko Kostadinov
Keyword(s):  
Linear Combination ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Special Cases ◽  
Sufficient And Necessary Conditions ◽  
Alpha Beta ◽  
Necessary And Sufficient

The aim of this paper is to provide sufficient and necessary conditions under which the linear combination $\alpha A + \beta B$, for given operators $A,B \in {\cal B}({\cal H})$ and $\alpha, \beta \in \mathbb{C}\setminus \lbrace 0 \rbrace$, is injective. Using these results, necessary and sufficient conditions for left (right) invertibility are given. Some special cases will be studied as well.

scholarly journals An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

2021 ◽  
Author(s):  
Yann Donnelly ◽  
Michael Peter Kennedy
Keyword(s):  
Nonlinear Distortion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantization Noise ◽  
Linear Combinations ◽  
Upper Limit ◽  
Comprehensive Theory ◽  
Free Behavior ◽  
Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

scholarly journals Linear combinations of polynomials with three-term recurrence

2021 ◽  
Vol 110 (124) ◽  
pp. 29-40
Author(s):  
Khang Tran ◽  
Maverick Zhang
Keyword(s):  
Linear Combination ◽  
Chebyshev Polynomials ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Combinations ◽  
Zero Distribution ◽  
Necessary And Sufficient ◽  
Linear Polynomials

We study the zero distribution of the sum of the first n polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of az + b for a, b ? R, of Chebyshev polynomials. In particular, we find necessary and sufficient conditions on a, b such that this linear combination is hyperbolic.

scholarly journals On nonsingularity and group invertibility of combinations of two group invertible matrices

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3675-3687
Author(s):  
Yu Li ◽  
Kezheng Zuo
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient ◽  
Invertible Matrices

Let A and B be two group invertible matrices, we study the rank, the nonsingularity and the group invertibility of A-B, AA#-BB#, c1A + c2B, c1A + c2B + c3AA#B where c1,c2 are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c1A + c2B + c3and c1A + c2B + c3+ c4BA to be nonsingular and group invertible are presented, which generalized some related results of Ben?tez, Liu, Koliha and Zuo [4, 17, 19, 25].

scholarly journals A Tauberian theorem for the statistical generalized Nörlund-Euler summability method

2019 ◽  
Vol 11 (2) ◽  
pp. 251-263
Author(s):  
Naim L. Braha
Keyword(s):  
Tauberian Theorem ◽  
Sufficient Conditions ◽  
Summability Method ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Necessary And Sufficient

Abstract Let (pn) and (qn) be any two non-negative real sequences with {{\rm{R}}_{\rm{n}}}: = \sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n}} - {\rm{k}}}}} \ne 0\,\,\,\,\left( {{\rm{n}} \in {\rm\mathbb{N}}} \right) With {\rm{E}}_{\rm{n}}^1 − we will denote the Euler summability method. Let (xn) be a sequence of real or complex numbers and set {\rm{N}}_{{\rm{p}},{\rm{q}}}^{\rm{n}}{\rm{E}}_{\rm{n}}^1: = {1 \over {{{\rm{R}}_{\rm{n}}}}}\sum\limits_{{\rm{k}} = 0}^{\rm{n}} {{{\rm{p}}_{\rm{k}}}{{\rm{q}}_{{\rm{n - k}}}}{1 \over {{2^{\rm{k}}}}}\sum\limits_{{\rm{v}} = 0}^{\rm{k}} {\left( {_{\rm{v}}^{\rm{k}}} \right){{\rm{x}}_{\rm{v}}}} } for n ∈ ℕ. In this paper, we present necessary and sufficient conditions under which the existence of the st− limit of (xn) follows from that of {\rm{st - N}}_{{\rm{p}},q}^{\rm{n}}{\rm{E}}_{\rm{n}}^1 − limit of (xn). These conditions are one-sided or two-sided if (xn) is a sequence of real or complex numbers, respectively.

scholarly journals The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring

2007 ◽  
Vol 83 (3) ◽  
pp. 423-438 ◽  
Author(s):  
Yaoming Yu ◽  
Guorong Wang
Keyword(s):  
Associative Ring ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Arbitrary Matrix ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Explicit Expressions

AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

Orthonormal expansions and the principle of uniform boundedness

1991 ◽  
Vol 43 (2) ◽  
pp. 341-347
Author(s):  
S.A. Husain ◽  
V.M. Sehgal
Keyword(s):  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Orthonormal Sequence ◽  
Necessary And Sufficient ◽  
Complex Valued ◽  
Orthonormal Expansions

Let {φν: ν ∈ N (non-negative integers)} ⊆ C[0, 1] be a complete orthonormal sequence of complex-valued functions in L2[0, 1], {λν: ν ∈ N} and {λνμ: ν, μ ∈ N} be sequences of complex numbers. In this paper, the necessary and sufficient conditions are developed for the series to converge and also to exist, in C[0, 1] for each f ∈ L1[0, 1] where .

scholarly journals An Exposition of Fractional Spurs Resulting From Nonlinear Distortion

2021 ◽  
Author(s):  
Yann Donnelly ◽  
Michael Peter Kennedy
Keyword(s):  
Nonlinear Distortion ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quantization Noise ◽  
Linear Combinations ◽  
Upper Limit ◽  
Comprehensive Theory ◽  
Free Behavior ◽  
Necessary And Sufficient

The interaction between quantization noise intro-duced by the divider controller and memoryless nonlinearities in a fractional-N PLL causes fractional spurs to occur. This paper presents a comprehensive theory to explain why combinations of quantizers and memoryless nonlinearities produce fractional spurs. Necessary and sufficient conditions for spur-free behavior in the presence of an arbitrary memoryless nonlinearity or linear combinations of sets of arbitrary memoryless nonlinearities are derived. Finally, an upper limit on the number of nonlinearities for which a quantizer can exhibit spur-free performance is derived.

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