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1
Content available remote

Generalized matrix functions and determinants

100%
Jafari M., Madadi A.
Open Mathematics
|
2014
|
tom 12
|
nr 3
464-469
EN
In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.
2
Content available remote

Determinant evaluations for binary circulant matrices

81%
Kravvaritis C.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
Determinant formulas for special binary circulant matrices are derived and a new open problem regarding the possible determinant values of these specific circulant matrices is stated. The ideas used for the proofs can be utilized to obtain more determinant formulas for other binary circulant matrices, too. The superiority of the proposed approach over the standard method for calculating the determinant of a general circulant matrix is demonstrated.
3
Content available remote

A new characterization of generalized complementary basic matrices

81%
Fiedler M., Hall F. J.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
In this paper, a new characterization of previously studied generalized complementary basic matrices is obtained. It is in terms of ranks and structure ranks of submatrices defined by certain diagonal positions. The results concern both the irreducible and general cases.
4
Content available remote

A counterexample to the Drury permanent conjecture

81%
Hutchinson G.
Special Matrices
|
2017
|
tom 5
|
nr 1
301-302
EN
We offer a counterexample to a conjecture concerning the permanent of positive semidefinite matrices. The counterexample is a 4 × 4 complex correlation matrix.
5
Content available remote

Bilinear characterizations of companion matrices

62%
Lin M., Wimmer H. K.
Special Matrices
|
2014
|
tom 2
|
nr 1
EN
Companion matrices of the second type are characterized by properties that involve bilinear maps.
6
Content available remote

Bounds for the Z-eigenpair of general nonnegative tensors

62%
Liu Q., Li Y.
Open Mathematics
|
2016
|
tom 14
|
nr 1
181-194
EN
In this paper, we consider the Z-eigenpair of a tensor. A lower bound and an upper bound for the Z-spectral radius of a weakly symmetric nonnegative irreducible tensor are presented. Furthermore, upper bounds of Z-spectral radius of nonnegative tensors and general tensors are given. The proposed bounds improve some existing ones. Numerical examples are reported to show the effectiveness of the proposed bounds.
7
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Determinants and inverses of circulant matrices with complex Fibonacci numbers

62%
Altınışık E., Feyza Yalçın N., Büyükköse Ş.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
Let ℱn = circ (︀F*1 , F*2, . . . , F*n︀ be the n×n circulant matrix associated with complex Fibonacci numbers F*1, F*2, . . . , F*n. In the present paper we calculate the determinant of ℱn in terms of complex Fibonacci numbers. Furthermore, we show that ℱn is invertible and obtain the entries of the inverse of ℱn in terms of complex Fibonacci numbers.
8
Content available remote

A note on the determinant of a Toeplitz-Hessenberg matrix

62%
Merca M.
Special Matrices
|
2013
|
tom 1
10-16
EN
The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.
9
Content available remote

On characteristic and permanent polynomials of a matrix

62%
Singh R., Bapat R. B.
Special Matrices
|
2017
|
tom 5
|
nr 1
97-112
EN
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.
10
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On the arrowhead-Fibonacci numbers

52%
Gültekin I., Deveci Ö.
Open Mathematics
|
2016
|
tom 14
|
nr 1
1104-1113
EN
In this paper, we define the arrowhead-Fibonacci numbers by using the arrowhead matrix of the characteristic polynomial of the k-step Fibonacci sequence and then we give some of their properties. Also, we study the arrowhead-Fibonacci sequence modulo m and we obtain the cyclic groups from the generating matrix of the arrowhead-Fibonacci numbers when read modulo m. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the arrowhead-Fibonacci sequence modulo m.
11
Content available remote

Another formulation of the Wick’s theorem. Farewell, pairing?

52%
Beloussov I. V.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
The algebraic formulation of Wick’s theorem that allows one to present the vacuum or thermal averages of the chronological product of an arbitrary number of field operators as a determinant (permanent) of the matrix is proposed. Each element of the matrix is the average of the chronological product of only two operators. This formulation is extremely convenient for practical calculations in quantum field theory, statistical physics, and quantum chemistry by the standard packages of the well known computer algebra systems.
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