Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithm

Skew circulant and circulant matrices have been an ideal research area and hot issue for solving various differential equations. In this paper, the skew circulant type matrices with the sum of Fibonacci and Lucas numbers are discussed. The invertibility of the skew circulant type matrices is considered. The determinant and the inverse matrices are presented. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, the maximum row sum matrix norm, and bounds for the spread of these matrices are given, respectively.

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On the spectral and Frobenius norm of a generalized Fibonacci r-circulant matrix

Special Matrices ◽

10.1515/spma-2018-0003 ◽

2018 ◽

Vol 6 (1) ◽

pp. 23-36

Author(s):

Jorma K. Merikoski ◽

Pentti Haukkanen ◽

Mika Mattila ◽

Timo Tossavainen

Keyword(s):

Lower Bounds ◽

Circulant Matrix ◽

Spectral Norm ◽

Frobenius Norm

Abstract Consider the recursion g0 = a, g1 = b, gn = gn−1 + gn−2, n = 2, 3, . . . . We compute the Frobenius norm of the r-circulant matrix corresponding to g0, . . . , gn−1. We also give three lower bounds (with equality conditions) for the spectral norm of this matrix. For this purpose, we present three ways to estimate the spectral norm from below in general.

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Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Abstract and Applied Analysis ◽

10.1155/2015/428146 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8

Author(s):

Wenai Xu ◽

Zhaolin Jiang

Keyword(s):

Kronecker Product ◽

Hadamard Product ◽

Circulant Matrix ◽

Spectral Norm ◽

Circulant Matrices ◽

Frobenius Norm

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

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The Difference between Optimal Rank‐1 Hankel Approximations in the Frobenius Norm and the Spectral Norm

PAMM ◽

10.1002/pamm.202000085 ◽

2021 ◽

Vol 20 (1) ◽

Author(s):

Hanna Knirsch ◽

Markus Petz ◽

Gerlind Plonka

Keyword(s):

Spectral Norm ◽

Frobenius Norm ◽

The Difference

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On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers

Abstract and Applied Analysis ◽

10.1155/2014/483021 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10 ◽

Cited By ~ 4

Author(s):

Zhaolin Jiang ◽

Jinjiang Yao ◽

Fuliang Lu

Keyword(s):

Fibonacci Numbers ◽

Spectral Norm ◽

Matrix Norm ◽

Circulant Matrices ◽

Frobenius Norm ◽

Transformation Matrices ◽

Skew Circulant

Circulant and skew circulant matrices have become an important tool in networks engineering. In this paper, we consider skew circulant type matrices with any continuous Fibonacci numbers. We discuss the invertibility of the skew circulant type matrices and present explicit determinants and inverse matrices of them by constructing the transformation matrices. Furthermore, the maximum column sum matrix norm, the spectral norm, the Euclidean (or Frobenius) norm, and the maximum row sum matrix norm and bounds for the spread of these matrices are given, respectively.

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The Bounds for Spectral Norm and Frobenius Norm Condition Number of a Simple Matrix

2011 Fourth International Conference on Information and Computing ◽

10.1109/icic.2011.123 ◽

2011 ◽

Author(s):

Xingdong Yang ◽

Cheng Chen ◽

Zhiying Ding ◽

Jiajing Zhang

Keyword(s):

Condition Number ◽

Spectral Norm ◽

Frobenius Norm ◽

Simple Matrix

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The core inverse and constrained matrix approximation problem

Open Mathematics ◽

10.1515/math-2020-0178 ◽

2020 ◽

Vol 18 (1) ◽

pp. 653-661 ◽

Cited By ~ 2

Author(s):

Hongxing Wang ◽

Xiaoyan Zhang

Keyword(s):

Unique Solution ◽

Approximation Problem ◽

Frobenius Norm ◽

Matrix Approximation ◽

The Core ◽

Core Inverse

Abstract In this article, we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse: ||Mx-b|{|}_{F}=\hspace{.25em}\min \hspace{1em}\text{subject}\hspace{.25em}\text{to}\hspace{1em}x\in {\mathcal R} (M), where M\in {{\mathbb{C}}}_{n}^{\text{CM}} . We get the unique solution to the problem, provide two Cramer’s rules for the unique solution and establish two new expressions for the core inverse.

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