Matrix Norm and Spectral Norm

Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

Abstract and Applied Analysis ◽

10.1155/2015/951340 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

Zhaolin Jiang ◽

Yunlan Wei

Keyword(s):

Differential Equations ◽

Research Area ◽

Spectral Norm ◽

Matrix Norm ◽

Circulant Matrices ◽

Frobenius Norm ◽

Lucas Numbers ◽

Skew Circulant ◽

Fibonacci And Lucas Numbers

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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Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with the Generalized Tribonacci Numbers

Earthline Journal of Mathematical Sciences ◽

10.34198/ejms.6121.131151 ◽

2021 ◽

pp. 131-151

Author(s):

Yüksel Soykan

Keyword(s):

Circulant Matrix ◽

Spectral Norm ◽

Matrix Norm ◽

Circulant Matrices ◽

Euclidean Norm ◽

Hadamard Products ◽

Tribonacci Numbers

In this paper, we obtain explicit Euclidean norm, eigenvalues, spectral norm and determinant of circulant matrix with the generalized Tribonacci (generalized (r, s, t)) numbers. We also present the sum of entries, the maximum column sum matrix norm and the maximum row sum matrix norm of this circulant matrix. Moreover, we give some bounds for the spectral norms of Kronecker and Hadamard products of circulant matrices of (r, s, t) and Lucas (r, s, t) numbers.

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On k-circulant matrices with the generalized third-order Pell numbers

Notes on Number Theory and Discrete Mathematics ◽

10.7546/nntdm.2021.27.4.187-206 ◽

2021 ◽

Vol 27 (4) ◽

pp. 187-206

Author(s):

Yüksel Soykan ◽

Keyword(s):

Circulant Matrix ◽

Numerical Results ◽

Spectral Norm ◽

Matrix Norm ◽

Circulant Matrices ◽

Euclidean Norm ◽

Third Order ◽

Pell Numbers

In this paper, we obtain explicit forms of the sum of entries, the maximum column sum matrix norm, the maximum row sum matrix norm, Euclidean norm, eigenvalues and determinant of k-circulant matrix with the generalized third-order Pell numbers. We also study the spectral norm of this k-circulant matrix. Furthermore, some numerical results for demonstrating the validity of the hypotheses of our results are given.

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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 Global Convergence of the Nonlinear Power Method for Mixed-Subordinate Matrix Norms

Journal of Scientific Computing ◽

10.1007/s10915-021-01524-w ◽

2021 ◽

Vol 88 (1) ◽

Author(s):

Antoine Gautier ◽

Matthias Hein ◽

Francesco Tudisco

Keyword(s):

Global Convergence ◽

Convergence Theorem ◽

Matrix Norm ◽

Nonnegative Matrices ◽

Np Hard ◽

Power Method ◽

Contraction Ratio ◽

Matrix Norms ◽

Vector Norms

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

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