Wiener-Hopf factorization of singular matrix functions

1998 ◽  
pp. 219-230
Author(s):  
M. Rakowski
Keyword(s):  
Matrix Functions ◽  
Singular Matrix

scholarly journals Rational Krylov methods for fractional diffusion problems on graphs

2021 ◽  
Author(s):  
Michele Benzi ◽  
Igor Simunec
Keyword(s):  
Analytic Function ◽  
Diffusion Equation ◽  
Graph Laplacian ◽  
Fractional Diffusion ◽  
Singular Matrix ◽  
Krylov Methods ◽  
Fractional Powers ◽  
Directed Networks ◽  
Rank One ◽  
Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

The semigroup generated by a unitary orbit of a singular matrix

2007 ◽  
Vol 55 (5) ◽  
pp. 417-428 ◽  
Author(s):  
H. Radjavi ◽  
A. R. Sourour
Keyword(s):  
Singular Matrix ◽  
Unitary Orbit

scholarly journals A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

1970 ◽  
Vol 11 (1) ◽  
pp. 81-83 ◽  
Author(s):  
Yik-Hoi Au-Yeung
Keyword(s):  
Complex Numbers ◽  
Single Element ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Simultaneous Diagonalization ◽  
Hermitian Matrices ◽  
Skew Field ◽  
Conjugate Transpose ◽  
Necessary And Sufficient ◽  
Real Quaternions

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

scholarly journals Very badly approximable matrix functions

2005 ◽  
Vol 11 (1) ◽  
pp. 127-154 ◽  
Author(s):  
V. V. Peller ◽  
S. R. Treil
Keyword(s):  
Matrix Functions ◽  
Badly Approximable

A New Random Data Description and Its Use in Transferring Road Roughness to Vehicle Response

1974 ◽  
Vol 96 (2) ◽  
pp. 676-679 ◽  
Author(s):  
J. C. Wambold ◽  
W. H. Park ◽  
R. G. Vashlishan
Keyword(s):  
Spectral Density ◽  
Frequency Distribution ◽  
Descriptive Analysis ◽  
Joint Probability ◽  
Amplitude Distribution ◽  
Matrix Functions ◽  
Initial Portion ◽  
Power Spectral ◽  
Road Roughness ◽  
Cross Spectral Density

The initial portion of the paper discusses the more conventional method of obtaining a vehicle transfer function where phase and magnitude are determined by dividing the cross spectral density of the input/output by the power spectral density (PSD) of the input. The authors needed a more descriptive analysis (over PSD) and developed a new signal description called Amplitude Frequency Distribution (AFD); a discrete joint probability of amplitude and frequency with the advantage of retaining amplitude distribution as well as frequency distribution. A better understanding was obtained, and transfer matrix functions were developed using AFD.

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