scholarly journals Karakteristik Matriks sebagai Daerah Asal Suatu Logaritma

2018 ◽  
Vol 6 (1) ◽  
pp. 60
Author(s):  
Era Dewi Kartika
Keyword(s):  
Positive Characteristic ◽  
Identity Matrix ◽  
Singular Matrix ◽  
Natural Logarithm ◽  
The Matrix ◽  
Characteristic Values ◽  
Commutative Matrix ◽  
Jordan Matrix ◽  
Logarithm Function ◽  
Square Matrix

Abstrak Rumus umum fungsi logaritma asli dengan daerah asal suatu matriks adalah ln⁡A=T S_((J_A ) ) {ln⁡〖(λ_1 I^((p_1 ) )+H^((p_1 ) ) ),ln⁡(λ_2 I^((p_2 ) )+H^((p_2 ) ) ),…,ln⁡(λ_u I^((p_u ) )+H^((p_u ) ) ) 〗 } 〖S_((J_A ) )〗^(-1) T^(-1) dengan T adalah matriks non-singular dimana A=TJ_A T^(-1), S_((J_A ) )adalah sebarang matriks yang komutatif dengan J_A, J_A adalah matriks Jordan dari matriks A, λ_i adalah nilai karakteristik dari pembagi elementer A, I adalah matriks identitas dan H^((p)) adalah matriks berukuran p×p yang mempunyai 1 sebagai anggota pada superdiagonal pertama dan 0 untuk lainnya. Karakteristik matriks A sebagai daerah asal suatu fungsi logaritma adalah matriks persegi yang non-singular dengan nilai-nilai karakteristik real positif Kata Kunci: matriks, daerah asal, logaritma asli Abstract The general formula of the natural logarithm function with domain of a matrix is ln⁡A=T S_((J_A ) ) {ln⁡〖(λ_1 I^((p_1 ) )+H^((p_1 ) ) ),ln⁡(λ_2 I^((p_2 ) )+H^((p_2 ) ) ),…,ln⁡(λ_u I^((p_u ) )+H^((p_u ) ) ) 〗 } 〖S_((J_A ) )〗^(-1) T^(-1) with T is the non-singular matrix which A=TJ_A T^(-1), S_((J_A ) ) is any commutative matrix with J_A, J_Ais the Jordan matrix of the matrix A, λ_i is the characteristic value of the elementary divider A, I is the identity matrix and H^((p)) is a square matrix which has 1 as a member of the first superdiagonal and 0 for other. The characteristic of matrix A as domain of a natural logarithm function is a non-singular square matrix with real positive characteristic values Keywords: matrix, domain, natural logarithm

On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)

1972 ◽  
Vol 15 (1) ◽  
pp. 45-49
Author(s):  
P. Basavappa
Keyword(s):  
Matrix Equation ◽  
Identity Matrix ◽  
Normal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Matrix Identities ◽  
Square Matrix

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.

scholarly journals Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2226
Author(s):  
Arif Mandangan ◽  
Hailiza Kamarulhaili ◽  
Muhammad Asyraf Asbullah
Keyword(s):  
Computer Graphics ◽  
Matrix Inversion ◽  
Floating Point ◽  
Large Integer ◽  
Identity Matrix ◽  
Singular Matrix ◽  
Integer Matrix ◽  
Unimodular Matrix ◽  
The Matrix ◽  
Lattice Based Cryptography

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.

scholarly journals On the Canonical Form of a Rational Integral Function of a Matrix

1932 ◽  
Vol 3 (2) ◽  
pp. 135-143 ◽  
Author(s):  
D. E. Rutherford
Keyword(s):  
Canonical Form ◽  
Integral Function ◽  
The Other ◽  
Singular Matrix ◽  
The Matrix ◽  
Image Position ◽  
Rational Integral ◽  
Square Matrix

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

scholarly journals Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂ n

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jorge Luis Arroyo Neri ◽  
Armando Sánchez-Nungaray ◽  
Mauricio Hernández Marroquin ◽  
Raquiel R. López-Martínez
Keyword(s):  
Toeplitz Operators ◽  
Complex Space ◽  
Fock Space ◽  
Identity Matrix ◽  
The Matrix

We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.

8. Matrices and groups

2015 ◽  
pp. 101-110
Author(s):  
Peter M. Higgins
Keyword(s):  
Dihedral Group ◽  
Inverse Matrix ◽  
Identity Matrix ◽  
Matrix Products ◽  
Square Matrix

‘Matrices and groups’ continues with the example of geometric matrix products to see what happens when we compose the mappings involved. It explains several features, including the identity matrix, the inverse matrix, the square matrix, and the concept of isomorphism. If a collection of matrices represent the elements of a group, such as the eight matrices that represent the dihedral group D, then each of these matrices A will have an inverse, A −1, such that AA-1 = A-1A =I, the identity matrix. This prompts the twin questions of when the inverse of a square matrix A exists and, if it does, how to find it.

scholarly journals A Numerical Method for Computing the Roots of Non-Singular Complex-Valued Matrices

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 966
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Numerical Method ◽  
Worked Examples ◽  
Higher Order ◽  
Singular Matrix ◽  
The Matrix ◽  
General Complex ◽  
Complex Valued

A method for the computation of the n th roots of a general complex-valued r × r non-singular matrix ? is presented. The proposed procedure is based on the Dunford–Taylor integral (also ascribed to Riesz–Fantappiè) and relies, only, on the knowledge of the invariants of the matrix, so circumventing the computation of the relevant eigenvalues. Several worked examples are illustrated to validate the developed algorithm in the case of higher order matrices.

scholarly journals A dynamical systems view of network centrality

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130835 ◽  
Author(s):  
Peter Grindrod ◽  
Desmond J. Higham
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Dynamic Networks ◽  
Network Evolution ◽  
Mathematical Framework ◽  
The Novel ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Systems View ◽  
Logarithm Function

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots , or timeslices , as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

scholarly journals Matrices of integers associated with self-transformations of surfaces

1948 ◽  
Vol 193 (1032) ◽  
pp. 25-43
Keyword(s):  
Rational Curve ◽  
Pell Equation ◽  
Quartic Surface ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
The Matrix ◽  
Quartic Surfaces ◽  
Infinite Sequences ◽  
Square Matrix

Let c = ( c 1 , . . . , c r ) be a set of curves forming a minimum base on a surface, which, under a self-transformation, T , of the surface, transforms into a set T c expressible by the equivalences T c = Tc, where T is a square matrix of integers. Further, let the numbers of common points of pairs of the curves, c i , c j , be written as a symmetrical square matrix Г. Then the matrix T satisfies the equation TГT' = Г. The significance of solutions of this equation for a given matrix Г is discussed, and the following special surfaces are investigated: §§4-7. Surfaces, in particular quartic surfaces, wìth only two base curves. Self-transformations of these depend on the solutions of the Pell equation u 2 - kv 2 = 1 (or 4). §8. The quartic surface specialized only by being made to contain a twisted cubic curve. This surface has an involutory transformation determined by chords of the cubic, and has only one other rational curve on it, namely, the transform of the cubic. The appropriate Pell equation is u 2 - 17 v 2 = 4. §9. The quartic surface specialized only by being made to contain a line and a rational curve of order m to which the line is ( m - 1)⋅secant (for m = 1 the surface is made to contain two skew lines). The surface has two infinite sequences of self-transformations, expressible in terms of two transformations R and S , namely, a sequence of involutory transformations R S n , and a sequence of non-involutory transformations S n .

Sign in / Sign up

Export Citation Format

Share Document