The canonical form of invariant matrices

1970 ◽  
Vol 68 (1) ◽  
pp. 21-25 ◽  
Author(s):  
D. B. Hunter
Keyword(s):  
Canonical Form ◽  
General Matrix ◽  
Invariant Matrices

1. Introduction. Let A[λ] be the irreducible invariant matrix of a general matrix of order n × n, corresponding to a partition (λ) = (λ1, λ2, …, λr) of some integer m. The problem to be discussed here is that of determining the canonical form of A[λ] when that of A is known.

Almost commutative matrices

1994 ◽  
Vol 445 (1925) ◽  
pp. 653-656
Keyword(s):  
Canonical Form ◽  
General Matrix

The n × n matrices A and X over a field F are called almost commutative if AX - XA = I . This equation cannot hold if the characteristic of F is either zero or greater than n . In the case where the characteristic of F divides n , certain pairs A and X , exist. It is the purpose of this paper not only to prove the existence of such pairs, but to construct (in terms of arbitrary parameters) the most general matrix X for a given matrix A . The methods use the rational canonical form so as to facilitate constructability.

scholarly journals Register-based implementation of the sparse general matrix-matrix multiplication on GPUs

2018 ◽  
Vol 53 (1) ◽  
pp. 407-408
Author(s):  
Junhong Liu ◽  
Xin He ◽  
Weifeng Liu ◽  
Guangming Tan
Keyword(s):  
Matrix Multiplication ◽  
General Matrix

scholarly journals Open associahedra and scattering forms

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Aidan Herderschee ◽  
Fei Teng
Keyword(s):  
Scattering Amplitudes ◽  
Canonical Form ◽  
Recursion Relations ◽  
Fiber Product ◽  
New Phenomena

Abstract We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.

scholarly journals New Approach to the Planetary Theory

1979 ◽  
Vol 81 ◽  
pp. 69-72 ◽  
Author(s):  
Manabu Yuasa ◽  
Gen'ichiro Hori
Keyword(s):  
Perturbation Method ◽  
Canonical Form ◽  
Equations Of Motion ◽  
Disturbing Function ◽  
Averaging Principle ◽  
The Sun ◽  
Planetary Theory ◽  
New Approach ◽  
Canonical Perturbation ◽  
Relative Coordinates

A new approach to the planetary theory is examined under the following procedure: 1) we use a canonical perturbation method based on the averaging principle; 2) we adopt Charlier's canonical relative coordinates fixed to the Sun, and the equations of motion of planets can be written in the canonical form; 3) we adopt some devices concerning the development of the disturbing function. Our development can be applied formally in the case of nearly intersecting orbits as the Neptune-Pluto system. Procedure 1) has been adopted by Message (1976).

scholarly journals Centrality-friendship paradoxes: when our friends are more important than us

2018 ◽  
Vol 7 (4) ◽  
pp. 515-528 ◽  
Author(s):  
Desmond J Higham
Keyword(s):  
Network Science ◽  
Sufficient Conditions ◽  
Sampling Bias ◽  
Average Degree ◽  
Science Literature ◽  
Eigenvector Centrality ◽  
General Matrix ◽  
Theoretical Support ◽  
The Social ◽  
Katz Centrality

Abstract The friendship paradox states that, on average, our friends have more friends than we do. In network terms, the average degree over the nodes can never exceed the average degree over the neighbours of nodes. This effect, which is a classic example of sampling bias, has attracted much attention in the social science and network science literature, with variations and extensions of the paradox being defined, tested and interpreted. Here, we show that a version of the paradox holds rigorously for eigenvector centrality: on average, our friends are more important than us. We then consider general matrix-function centrality, including Katz centrality, and give sufficient conditions for the paradox to hold. We also discuss which results can be generalized to the cases of directed and weighted edges. In this way, we add theoretical support for a field that has largely been evolving through empirical testing.

A QR-type reduction for computing the SVD of a general matrix product/quotient

2003 ◽  
Vol 95 (1) ◽  
pp. 101-121 ◽  
Author(s):  
Delin Chu ◽  
Lieven De Lathauwer ◽  
Bart De Moor
Keyword(s):  
Matrix Product ◽  
General Matrix ◽  
Type Reduction
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