On the inverse sum indeg energy of trees

Let [Formula: see text] be a graph of order [Formula: see text] and [Formula: see text] be the degree of the vertex [Formula: see text], for [Formula: see text]. The [Formula: see text] matrix of [Formula: see text] is the square matrix of order [Formula: see text] whose [Formula: see text]-entry is equal to [Formula: see text] if [Formula: see text] is adjacent to [Formula: see text], and zero otherwise. Let [Formula: see text], be the eigenvalues of [Formula: see text] matrix. The [Formula: see text] energy of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of the absolute values of the eigenvalues of [Formula: see text] matrix. In this paper, we prove that the star has the minimum [Formula: see text] energy among trees.

The algebra generated by nilpotent elements in a matrix centralizer

For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$

Determinan Matriks Sirkulan Dengan Metode Kondensasi Dodgson

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.

MASALAH EIGEN DAN EIGENMODE MATRIKS ATAS ALJABAR MIN-PLUS

Eigen problems and eigenmode are important components related to square matrices. In max-plus algebra, a square matrix can be represented in the form of a graph called a communication graph. The communication graph can be strongly connected graph and a not strongly connected graph. The representation matrix of a strongly connected graph is called an irreducible matrix, while the representation matrix of a graph that is not strongly connected is called a reduced matrix. The purpose of this research is set the steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and also eigenmode of the regular reduced matrix over min-plus algebra. Min-plus algebra has an ispmorphic structure with max-plus algebra. Therefore, eigen problems and eigenmode matrices over min-plus algebra can be determined based on the theory of eigenvalues, eigenvectors and eigenmode matrices over max-plus algebra. The results of this research obtained steps to determine the eigenvalues and eigenvectors of the irreducible matrix over min-plus algebra and eigenmode algorithm of the regular reduced matrix over min-plus algebra

Some Rank Formulas for the Yang-Baxter Matrix Equation AXA = XAX

Let A be an arbitrary square matrix, then equation AXA =XAX with unknown X is called Yang-Baxter matrix equation. It is difficult to find all the solutions of this equation for any given A . In this paper, the relations between the matrices A and B are established via solving the associated rank optimization problem, where B = AXA = XAX , and some analytical formulas are derived, which may be useful for finding all the solutions and analyzing the structures of the solutions of the above Yang-Baxter matrix equation.

A note on two-sided removal and cancellation properties associated with Hermitian matrix

A complex square matrix A is said to be Hermitian if A = A∗, the conjugate transpose of A. We prove that each of the two triple matrix product equalities AA∗A = A∗AA∗ and A3 = AA∗A implies that A is Hermitian by means of decompositions and determinants of matrices, which are named as two-sided removal and cancellation laws associated with a Hermitian matrix. We also present several general removal and cancellation laws as the extensions of the preceding two facts about Hermitian matrix.AMS classifications: 15A24, 15B57

Mixed Hill Cipher methods with triple pass protocol methods

Hill Cipher is a reimbursement coding system that converts specific textual content codes into numbers and does no longer exchange the location of fixed symbols. The symbol modifications simplest in step with the English letter table inclusive of (26) characters handiest. An encoded Hill Cipher algorithm was used that multiplication the square matrix of the apparent text with a non-public key and then combined it with the Triple Pass Protocol method used to repeat the encryption three times without relying on a personal key. Also, you could decode the code and go back it to the express textual content. The cause of mixing algorithms is to cozy the message without any key change among the sender and the recipient.

On Some Decompositions of Matrices over Algebraically Closed and Finite Fields

We study when every square matrix over an algebraically closed field or over a finite field is decomposable into a sum of a potent matrix and a nilpotent matrix of order 2. This can be related to our recent paper, published in Linear & Multilinear Algebra (2022). We also completely address the question when each square matrix over an infinite field can be decomposed into a periodic matrix and a nilpotent matrix of order 2

Building matrixes of higher order to achieve the special commutative multiplication and its applications in cryptography: بناء مصفوفات تبديلية من مراتب عليا وتطبيقاتها في التشفير

In this paper, we introduce a method for building matrices that verify the commutative property of multiplication on the basis of circular matrices, as each of these matrices can be divided into four circular matrices, and we can also build matrices that verify the commutative property of multiplication from higher order and are not necessarily divided into circular matrices. Using these matrixes, we provide a way to securely exchange a secret encryption key, which is a square matrix, over open communication channels, and then use this key to exchange encrypted messages between two sides or two parties. Moreover, using these matrixes we also offer a public-key encryption method, whereby the two parties exchange encrypted messages without previously agreeing on a common secret key between them.