scholarly journals On k-circulant matrices with arithmetic sequence

Filomat ◽  
2017 ◽  
Vol 31 (8) ◽  
pp. 2517-2525 ◽  
Author(s):  
Biljana Radicic
Keyword(s):  
Complex Number ◽  
Circulant Matrices ◽  
Arithmetic Sequence ◽  
Nonzero Complex Number ◽  
Singular Matrices ◽  
Invertible Matrices ◽  
The Way

Let k be a nonzero complex number. In this paper we consider k-circulant matrices with arithmetic sequence and investigate the eigenvalues, determinants and Euclidean norms of such matrices. Also, for k = 1, the inverses of such (invertible) matrices are obtained (in a way different from the way presented in [1]), and the Moore-Penrose inverses of such (singular) matrices are derived.

scholarly journals On k-circulant matrices with the Lucas numbers

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 4037-4046
Author(s):  
Biljana Radicic
Keyword(s):  
Complex Number ◽  
Circulant Matrix ◽  
Spectral Norm ◽  
Circulant Matrices ◽  
Euclidean Norm ◽  
Lucas Numbers ◽  
Lucas Number ◽  
Nonzero Complex Number

Let k be a nonzero complex number. In this paper, we determine the eigenvalues of a k-circulant matrix whose first row is (L1,L2,..., Ln), where Ln is the nth Lucas number, and improve the result which can be obtained from the result of Theorem 7. [28]. The Euclidean norm of such matrix is obtained. Bounds for the spectral norm of a k-circulant matrix whose first row is (L-11, L-12,..., L-1n ) are also investigated. The obtained results are illustrated by examples.

scholarly journals Determinants of binomial-related circulant matrices

2018 ◽  
Vol 6 (1) ◽  
pp. 262-272
Author(s):  
Trairat Jantaramas ◽  
Somphong Jitman ◽  
Pornpan Kaewsaard
Keyword(s):  
Positive Integer ◽  
Complex Number ◽  
Circulant Matrices ◽  
Algebraic Structures ◽  
Explicit Formulas ◽  
Open Problems ◽  
Left And Right ◽  
Nonzero Complex Number ◽  
Special Case

Abstract Due to their rich algebraic structures and wide applications, circulant matrices have been of interest and continuously studied. In this paper, n×n complex left and right circulant matrices whose first row consists of the coefficients in the expansion of (x + zy)n−1 are focused on, where z is a nonzero complex number and n is a positive integer. In the case where z ∈ {1, −1, i, −i}, explicit formulas for the determinants of such matrices are completely determined. Known results on the determinants of binomial circulant matrices can be viewed as the special case where z = 1. Finally, some remarks and open problems are discussed.

scholarly journals On the structure of least common multiple matrices from some class of matrices

2018 ◽  
Vol 10 (1) ◽  
pp. 179-184
Author(s):  
A.M. Romaniv
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Smith Normal Form ◽  
Least Common Multiple ◽  
The Matrix ◽  
Singular Matrices ◽  
Class Of Matrices ◽  
Smith Normal Forms ◽  
Invertible Matrices ◽  
Principal Ideal Domains

For non-singular matrices with some restrictions, we establish the relationships between Smith normal forms and transforming matrices (a invertible matrices that transform the matrix to its Smith normal form) of two matrices with corresponding matrices of their least common right multiple over a commutative principal ideal domains. Thus, for such a class of matrices, given answer to the well-known task of M. Newman. Moreover, for such matrices, received a new method for finding their least common right multiple which is based on the search for its Smith normal form and transforming matrices.

scholarly journals Massively Parallel Searching for Better Algorithms or How to Do a Cross Product with Five Multiplications

1996 ◽  
Vol 5 (3) ◽  
pp. 203-217
Author(s):  
John Gustafson ◽  
Srinivas Aluru
Keyword(s):  
General Theory ◽  
Computer Program ◽  
Complex Number ◽  
Cross Product ◽  
Massively Parallel ◽  
Matrix Product ◽  
Massive Parallelism ◽  
The Way

A number of "tricks" are known that trade multiplications for additions. The term "tricks" reflects the way these methods seem not to proceed from any general theory, but instead jump into existence as recipes that work. The Strassen method for 2 × 2 matrix product with seven multiplications is a well-known example, as is the method for finding a complex number product in three multiplications. We have created a practical computer program for finding such tricks automatically, where massive parallelism makes the combinatorially explosive search tolerable for small problems. One result of this program is a method for cross products of three-vectors that requires only five multiplications.

scholarly journals Expressions for the g-Drazin inverse in a Banach algebra

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3845-3854
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani
Keyword(s):  
Banach Algebra ◽  
Complex Number ◽  
Block Matrix ◽  
Explicit Representation ◽  
Drazin Inverse ◽  
Operator Matrices ◽  
Nonzero Complex Number ◽  
Generalized Drazin Inverse

We explore the generalized Drazin inverse in a Banach algebra. Let A be a Banach algebra, and let a,b ? Ad. If ab = ?a?bab? for a nonzero complex number ?, then a + b ? Ad. The explicit representation of (a + b)d is presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., 2015, 156934.8] are extended.

scholarly journals The ring of invariants of matrices

1986 ◽  
Vol 104 ◽  
pp. 149-161 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Number Field ◽  
Complex Number ◽  
Rational Representation ◽  
Invertible Matrices ◽  
Complex Number Field

We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n)i be the vector space of l-tuples of n × ra-matrices. We denote by ρ: G → GL(W) a rational representation of G defined as follows:if S ∈ G, A(i) ∈ M(n) (i = 1, 2, …, l).

ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

2016 ◽  
Vol 94 (2) ◽  
pp. 245-253 ◽  
Author(s):  
TOUFIK ZAÏMI
Keyword(s):  
Finite Number ◽  
Algebraic Number ◽  
Complex Number ◽  
Convergent Sequence ◽  
Pisot Number ◽  
Pisot Numbers ◽  
Limit Points ◽  
Nonzero Complex Number

We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

2014 ◽  
Vol 98 (3) ◽  
pp. 289-310 ◽  
Author(s):  
PETER BUNDSCHUH ◽  
KEIJO VÄÄNÄNEN
Keyword(s):  
Rational Function ◽  
Complex Number ◽  
Functional Equations ◽  
Algebraic Independence ◽  
Lucas Numbers ◽  
Mahler Functions ◽  
Independence Results ◽  
Nonzero Complex Number ◽  
Fibonacci And Lucas Numbers ◽  
Reciprocal Sums

This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

scholarly journals SCHUR MULTIPLICATIVE MAPS ON MATRICES

2008 ◽  
Vol 77 (1) ◽  
pp. 49-72 ◽  
Author(s):  
SEAN CLARK ◽  
CHI-KWONG LI ◽  
ASHWIN RASTOGI
Keyword(s):  
Symmetric Functions ◽  
Rank Function ◽  
Elementary Symmetric Functions ◽  
Linear Maps ◽  
Singular Matrices ◽  
Invertible Matrices ◽  
Determinant Function

AbstractThe structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying $f(S) \subseteq S$ for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f(A))=Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

On the Skitovich–Darmois theorem for complex and quaternion random variables

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Gennadiy Feldman
Keyword(s):  
Complex Number ◽  
Random Variables ◽  
Wide Sense ◽  
Narrow Sense ◽  
Gaussian Random Variables ◽  
Linear Forms ◽  
Nonzero Complex Number

AbstractWe prove the following theorem. Let {\alpha=a+ib} be a nonzero complex number. Then the following statements hold: (i) Let either {b\neq 0} or {b=0} and {a>0}. Let {\xi_{1}} and {\xi_{2}} be independent complex random variables. Assume that the linear forms {L_{1}=\xi_{1}+\xi_{2}} and {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent. Then {\xi_{j}} are degenerate random variables. (ii) Let {b=0} and {a<0}. Then there exist complex Gaussian random variables in the wide sense {\xi_{1}} and {\xi_{2}} such that they are not complex Gaussian random variables in the narrow sense, whereas the linear forms {L_{1}=\xi_{1}+\xi_{2}} and {L_{2}=\xi_{1}+\alpha\xi_{2}} are independent.

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