Certain Properties of Square Matrices over Fields with Applications to Rings

We prove that any square nilpotent matrix over a field is a difference of two idempotent matrices as well as that any square matrix over an algebraically closed field is a sum of a nilpotent square-zero matrix and a diagonalizable matrix. We further apply these two assertions to a variation of π-regular rings. These results somewhat improve on establishments due to Breaz from Linear Algebra & amp; Appl. (2018) and Abyzov from Siberian Math. J. (2019) as well as they also refine two recent achievements due to the present author, published in Vest. St. Petersburg Univ. - Ser. Math., Mech. & amp; Astr. (2019) and Chebyshevskii Sb. (2019), respectively.

A LINEAR INVARIANT OF A MATRIX RING OF SIZE TWO OVER A POLYNOMIAL RING

We consider a matrix ring of order two over a ring of polynomials in two variables with coefficients from a commutative associative integrity domain with unity. A linear mapping of this ring into the polynomial ring is presented, depending on a matrix of a special form, whose square is zero matrix. The value of this map is invariant with respect to conjugation by an invertible matrix of elements of the ring, including the matrix by which the map is constructed. The properties of the mapping thus obtained are described.

Algorithm for generating matrices of estimated effects for breeding evaluation of small cattle using a mixed biometric model

The use of a mixed biometric model for breeding evaluation of small cattle has been discussed in the article. This model of breeding evaluation involves a large number of matrix operations. At the same time, the volumes of the formed matrices are directly proportional to the number of animals in the evaluated sample as well as to the number of their off spring. An algorithm for generating matrices of estimated effects that have a large dimension has been presented in the paper. This task is the most time-consuming when using a mixed biometric model. Currently, there are the large number of mathematical packages that provide ample opportunities for performing calculations. A special place in this series is occupied by the integrated mathematical package MATLAB has been designed specifically for performing matrix operations. The authors rely on the use of this package in their work. At the same time the algorithm presented in this paper has the property of universality and can be applied by users in any other software product. Since the matrices of the estimated effects consist of zeros and ones we propose the two-step procedure for forming these matrices. At the first stage, a zero matrix of the required dimension is created. At the second stage, in accordance with the data on the number of evaluated animals, the number of herds for which off spring are distributed, the number and affiliation of evaluated animals to genetic groups, the elements of the matrix are determined, in which zeros are replaced by ones. The advantage of the proposed algorithm is its versatility, and the representation of the algorithm in the form of a block diagram will allow you to design it as a separate proceduresubroutine.

Zero Product of Three Two Level Toeplitz Operators

In this paper we investigate conditions for T_f1 T_f2 T_f3 - T_f1f2f3 = 0 where T_f1 , T_f2 , and T_f3 are bi-level Toeplitz operators on the Hardy space of bidisk and f_1; f_2; f_3 are bounded and measurable complex valued functions on bidisk. We also provide that T_f1 T_f2 T_f3 identical to zero matrix if and only if at least one of f_i is identically zero for 1 ≤i ≤3.

Cryptography based on the Matrices

In this work we introduce a new method of cryptography based on the matrices over a finite field $\mathbb{F}_{q}$, were $q$ is a power of a prime number $p$. The first time we construct thematrix $M=\left(\begin{array}{cc}A_{1} & A_{2} \\0 & A_{3} \\\end{array}\right)$ were \ $A_{i}$ \ with $i \in \{1, 2, 3 \}$ is the matrix oforder $n$ \ in \ $\mathcal{M}(\mathbb{F}_{q})$ - the set ofmatrices with coefficients in $\mathbb{F}_{q}$ - and $0$ is the zero matrix of order $n$. We prove that $M^{l}=\left(\begin{array}{cc}A_{1}^{l} & (A_{2})_{l} \\0 & A_{3}^{l} \\\end{array}\right)$ were $(A_{2})_{l}=\sum\limits_{k=0}^{l-1}A_{1}^{l-1-k}A_{2}A_{3}^{k}$ for all $l\in \mathbb{N}^{\ast}$. After we will make a cryptographic scheme between the two traditional entities Alice and Bob.

Zero-dilation Index of S_n-matrix and Companion Matrix

The zero-dilation index $d(A)$ of a square matrix $A$ is the largest $k$ for which $A$ is unitarily similar to a matrix of the form ${\scriptsize\left[\begin{array}{cc} 0_k & \ast\\ \ast & \ast\end{array}\right]}$, where $0_k$ denotes the $k$-by-$k$ zero matrix. In this paper, it is shown that if $A$ is an $S_n$-matrix or an $n$-by-$n$ companion matrix, then $d(A)$ is at most $\lceil n/2\rceil$, the smallest integer greater than or equal to $n/2$. Those $A$'s for which the upper bound is attained are also characterized. Among other things, it is shown that, for an odd $n$, the $S_n$-matrix $A$ is such that $d(A)=(n+1)/2$ if and only if $A$ is unitarily similar to $-A$, and, for an even $n$, every $n$-by-$n$ companion matrix $A$ has $d(A)$ equal to $n/2$

Nonlocal Cauchy problems for fractional order nonlinear differential systems

In this paper, we discuss nonlocal Cauchy problems for fractional order nonlinear differential systems. Firstly, an important matrix associated with fractional order and two functionals are constructed. Further, some sufficient conditions which guarantee such matrix convergent to zero matrix are presented. Secondly, by using three fixed point theorems via the techniques that use convergent to zero matrix and vector norm, some existence results for the solutions of such fractional order nonlinear differential systems are given under different conditions. Finally, some examples are given to illustrate the results.