scholarly journals Determinan Matriks Sirkulan Dengan Metode Kondensasi Dodgson

2021 ◽  
Vol 18 (2) ◽  
pp. 211-220
Author(s):  
M R Fahlevi
Keyword(s):  
Toeplitz Matrix ◽  
Matrix Method ◽  
Matrix Theory ◽  
Circulant Matrix ◽  
Original Matrix ◽  
Mental Computation ◽  
Condensation Method ◽  
The Matrix ◽  
Dodgson Condensation ◽  
Square Matrix

One of the important topics in mathematics is matrix theory. There are various types of matrix, one of which is a circulant matrix. Circulant matrix generally fulfill the same operating axioms as square matrix, except that there are some specific properties for the circulant matrix. Every square matrix has a determinant. The concept of determinants is very useful in the development of mathematics and across disciplines. One method of determining the determinant is condensation. The condensation method is classified as a method that is not widely known. The condensation matrix method in determining the determinant was proposed by several scientists, one of which was Charles Lutwidge Dodgson with the Dodgson condensation method. This paper will discuss the Dodgson condensation method in determining the determinant of the circulant matrix. The result of the condensation of the matrix will affect the size of the original matrix as well as new matrix entries. Changes in the circulant matrix after Dodgson's conduction load the Toeplitz matrix, in certain cases, the determinant of the circulant matrix can also be determined by simple mental computation.

scholarly journals On characteristic and permanent polynomials of a matrix

2017 ◽  
Vol 5 (1) ◽  
pp. 97-112 ◽  
Author(s):  
Ranveer Singh ◽  
R. B. Bapat
Keyword(s):  
Recurrence Relation ◽  
Matrix Theory ◽  
Open Problems ◽  
The Matrix ◽  
Laplace Expansion ◽  
Vertex Disjoint ◽  
Square Matrix

Abstract There is a digraph corresponding to every square matrix over ℂ. We generate a recurrence relation using the Laplace expansion to calculate the characteristic and the permanent polynomials of a square matrix. Solving this recurrence relation, we found that the characteristic and the permanent polynomials can be calculated in terms of the characteristic and the permanent polynomials of some specific induced subdigraphs of blocks in the digraph, respectively. Interestingly, these induced subdigraphs are vertex-disjoint and they partition the digraph. Similar to the characteristic and the permanent polynomials; the determinant and the permanent can also be calculated. Therefore, this article provides a combinatorial meaning of these useful quantities of the matrix theory. We conclude this article with a number of open problems which may be attempted for further research in this direction.

The relation of Eddington’s E-numbers to the tensor calculus - I. The matrix form of E-numbers

1952 ◽  
Vol 214 (1118) ◽  
pp. 292-311 ◽  
Keyword(s):  
Commutative Algebra ◽  
Matrix Theory ◽  
Matrix Transformations ◽  
Mathematical Form ◽  
Tensor Calculus ◽  
Square Roots ◽  
Physical Magnitude ◽  
Tensor Theory ◽  
The Matrix ◽  
Square Matrix

An E-number is an unusual mathematical form, constructed from sixteen algebraic variables and sixteen E-symbols (square roots of — 1 obeying a non-commutative algebra), which has distinctive properties different from those of the square matrix of the variables employed in tensor theory. When the latter is considered to be a tensor, the corresponding E-number cannot be a standard tensor, and tensor calculus has made no use of it. Without loss of generality the E-symbols can always be envisaged as transforms of a certain set of numerical matrices, and it is shown here that when this is done every E-number can be expressed in a simplest form, called an E -number. The reduction of an E- to an E -number takes the form of a transformation analogous to that by which a physical magnitude is expressed in a different unit, the variables and the symbols being transformed oppositely. When evaluated the E -number shows itself as a square matrix, which is characterized by having the data expressed by the sixteen variables arranged in another systematic way than that employed in the standard matrix of the variables. It may be obtained from this by a novel kind of transposition, in which the transposed elements are not the matrix terms themselves, but the symmetric and anti-symmetric parts of the terms. Alternatively, it may be obtained by submitting the standard matrix of the variables to a special type of trans­formation. The transformations employed in these operations differ from those normally used in matrix theory, as they take the form of sums of sets of matrix transformations. General formulae for the products of E -numbers are derived.

scholarly journals The quantum theory of dispersion

1927 ◽  
Vol 114 (769) ◽  
pp. 710-728 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Classical Theory ◽  
Matrix Theory ◽  
Hamiltonian Function ◽  
Spectral Lines ◽  
Light Quantum ◽  
Original Matrix ◽  
The Matrix ◽  
Points Of View

The new quantum mechanics could at first be used to answer questions concerning radiation only through analogies with the classical theory. In Heisenberg’s original matrix theory, for instance, it is assumed that the matrix elements of the polarisation of an atom determine the emission and absorption of radiation analogously to the Fourier components in the classical theory. In more recent theories a certain expression for the electric density obtained from the quantum mechanics is used to determine the emitted radiation by the same formulæ as in the classical theory. These methods give satisfactory results in many cases, but cannot even be applied to problems where the classical analogies are obscure or non-existent, such as resonance radiation and the breadths of spectral lines. A theory of radiation has been given by the author which rests on a more definite basis. It appears that one can treat a field of radiation as a dynamical system, whose interaction with an ordinary atomic system may be described by a Hamiltonian function. The dynamical variables specifying the field are the energies and phases of its various harmonic conrponents, each of which is effectively a simple harmonic oscillator. One must, of course, in the quantum theory take these variables to be q-numbers satisfying the proper quantum conditions. One finds then that the Hamiltonian for the interaction of the field with an atom is of the same form as that for the interaction of an assembly of light-quanta with the atom. There is thus a complete formal reconciliation between the wave and light-quantum points of view.

scholarly journals Random Toeplitz matrices: The condition number under high stochastic dependence

2021 ◽  
Author(s):  
Paulo Manrique-Mirón
Keyword(s):  
Generating Function ◽  
Condition Number ◽  
Toeplitz Matrix ◽  
Singular Values ◽  
Moment Generating Function ◽  
Circulant Matrix ◽  
Singular Value ◽  
The Matrix ◽  
Matrix Formula ◽  
The Moment

In this paper, we study the condition number of a random Toeplitz matrix. As a Toeplitz matrix is a diagonal constant matrix, its rows or columns cannot be stochastically independent. This situation does not permit us to use the classic strategies to analyze its minimum singular value when all the entries of a random matrix are stochastically independent. Using a circulant embedding as a decoupling technique, we break the stochastic dependence of the structure of the Toeplitz matrix and reduce the problem to analyze the extreme singular values of a random circulant matrix. A circulant matrix is, in fact, a particular case of a Toeplitz matrix, but with a more specific structure, where it is possible to obtain explicit formulas for its eigenvalues and also for its singular values. Among our results, we show the condition number of a non-symmetric random circulant matrix [Formula: see text] of dimension [Formula: see text] under the existence of the moment generating function of the random entries is [Formula: see text] with probability [Formula: see text] for any [Formula: see text], [Formula: see text]. Moreover, if the random entries only have the second moment, the condition number satisfies [Formula: see text] with probability [Formula: see text]. Also, we analyze the condition number of a random symmetric circulant matrix [Formula: see text]. For the condition number of a random (non-symmetric or symmetric) Toeplitz matrix [Formula: see text] we establish [Formula: see text], where [Formula: see text] is the minimum singular value of the matrix [Formula: see text]. The matrix [Formula: see text] is a random circulant matrix and [Formula: see text], where [Formula: see text] are deterministic matrices, [Formula: see text] indicates the conjugate transpose of [Formula: see text] and [Formula: see text] are random diagonal matrices. From random experiments, we conjecture that [Formula: see text] is well-conditioned if the moment generating function of the random entries of [Formula: see text] exists.

scholarly journals Partial Component Consensus of Discrete-Time Multiagent Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Wenjun Hu ◽  
Gang Zhang ◽  
Zhongjun Ma ◽  
Binbin Wu
Keyword(s):  
Multiagent Systems ◽  
Discrete Time ◽  
Matrix Theory ◽  
Pinning Control ◽  
Directed Network ◽  
Strong Function ◽  
Partial Stability ◽  
Partial Component ◽  
The Matrix ◽  
Error System

The multiagent system has the advantages of simple structure, strong function, and cost saving, which has received wide attention from different fields. Consensus is the most basic problem in multiagent systems. In this paper, firstly, the problem of partial component consensus in the first-order linear discrete-time multiagent systems with the directed network topology is discussed. Via designing an appropriate pinning control protocol, the corresponding error system is analyzed by using the matrix theory and the partial stability theory. Secondly, a sufficient condition is given to realize partial component consensus in multiagent systems. Finally, the numerical simulations are given to illustrate the theoretical results.

A Real-Valued Toeplitz Matrix Method for DOA Estimation

2020 ◽  
Vol 43 (4) ◽  
pp. 350-356
Author(s):  
Jianxiong Li ◽  
Deming Li ◽  
Xianguo Li
Keyword(s):  
Toeplitz Matrix ◽  
Matrix Method ◽  
Doa Estimation

scholarly journals MATRIX THEORY, HILBERT SCHEME AND INTEGRABLE SYSTEM

1998 ◽  
Vol 13 (34) ◽  
pp. 2731-2742 ◽  
Author(s):  
YUTAKA MATSUO
Keyword(s):  
Hilbert Scheme ◽  
Matrix Theory ◽  
Fock Space ◽  
Homology Class ◽  
Orthogonal Basis ◽  
Inner Product ◽  
Virasoro Symmetry ◽  
Hilbert Scheme Of Points ◽  
The Matrix ◽  
Boson Fock Space

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relates the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These bases can be related to the eigenfunctions of Calogero–Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

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