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4INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 1 ISSUE 5 UIF-2022: 8.2 | ISSN: 2181-3337
NILPOTENTS ZERO DIVISORS OF A MULTIDIMENSIONAL MATRIX
Masharipov Sirojiddin
National university of uzbekistan (phd) (doktorant) https://doi.org/10.5281/zenodo.7069796 Abstract. In this article, the necessary conditions for nilpotency of matrices of three and higher dimensions are studied. In addition, its application to quadratic stochastic operators is presented.
Keywords: nilpotent, quadratic stochastic operators, zero divisor, multidimensional matrix, simplex.
НИЛЬПОТЕНТЫ ДЕЛИТЕЛИ НУЛЯ МНОГОМЕРНЫХ МАТРИЦЫ Аннотация. В данной статье изучаются необходимые условия нильпотентной матриц трех и более размерностей. Кроме того, представлено его приложение к квадратичным стохастическим операторам.
Ключевые слова: нильпотент, квадратическое стохастическое операторы, делители нуля, многомерная матрица, симплекс.
INTRODUCTION
We know that quadratic stochastic operators depend on p k coefficients, that is way we
need to study three-dimensional matrices. Although matrices have been widely studied in n = 2, but the properties of matrices in three and higher dimensions have not been fully studied. This article is devoted to it and shows its connection with quadratic stochastic operators. MATERIALS AND METHODS
We introduce the necessary concepts of multidimensional matrices and quadratic stochastic operators. We mainly need three-dimensional matrices, so we give their definitions below.
Definition. Let T, T, TT be k x n matrices with real entries. A 3-d matrix A is
1 7 2 7 J 7 7 m
an array of the matrices T, T, T,•••, Tm arrayed from top to bottom. We denote this as A = [T | T I •••! Tm\ and call this the top representation of A . We also write A as follows. [1]
(1)
A is a m x k x n matrix.
Definition. The nonlinear stochastic operators are defined on a simplex S, and the dimensional of the simplex is (m — l), as [3]
m
Sm—l = {x. = (*2,X,,...,^n)e Rm,XX = l, and x. > 0}
¿=1
Definition. The evaluation of the nonlinear stochastic operators as
ai,u aU,2 ■ a,, 1,1,и a ,, /72,1,1 a 19 . /72,1,2 . a , /77,1,/2
A = • • • • • •
_ ai,k ,1 a\,k,2 • a,, 1 ,k,n a ,, m,k,\ a , 9 . m,k, 2 . a , m,k,n T
i, J=1
i x X ■
k i j
INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 1 ISSUE 5 UIF-2022: 8.2 | ISSN: 2181-3337
Where the p k is the transaction matrix under the condition:
m
o,Y%fc = i.
k=1
We write the operator in the form of a matrix. []
Pij,k — Pji,k ^ 0, ^ Pij,k —
. x =
( xl x2
( xj x2
( xj x2
x )
m s
x)
m
x)
m
a
& ,, V mi,i
a
a
V mi,2
a
a
V mi,m
,. ¿7,
¿7
.. ¿7,
a
a,
¿7
mm,m
Xo
VXm 7
A'0
x
V m y
A'0
x
V m //
(2)
If we multiply and calculate this matrix, we can write each in the following form.
V(Xj) = aniXiXi + a12,ixix2 h ••• h aim,\X\Xm h a21,1X2X1 h a22,1X2X2
^h... ^h i xx ^h... ^h a 11 x Xi ^h a rs 1 x Xo ^h ••• ^h a 1 x x
2 m,1 2 m m1,1 m 1 m 2,1 m 2 mm,1 m m
V ( x ) =
V(x2) = &ii,2xixi h &22,2xix2 h ••• h &im,2xixm h &2i,2x2xi h &22,2x2x2
^h••• ^h ^xox ^h ••• ^h & ^ ^x xi ^h & * ^x x^ ^h ••• ^h ^x x
2m,2 2 m mi,2 m i m 2,2 m 2 mm,2 m m
:=:+:+:+:+:+:+:+:+:+:+:+:+:+:+:+:+:+:+: V ( x ) ^ ^^ x^ x^ + x^x^ + ••• + x^x + x0x, + x0x0
V m/ ii,m i i i2,m i 2 im,m i m 2i,m 2 i 22,m 2 2
(3)
22, m 2 2
. ^h... ^h X ^h ... ^h X Xi ^h ^ X X/-* X X
2 m,m 2 m ml,m m 1 m 2,m m 2 mm,m mm
In our work, it is clear that a nonlinear operator can be constructed using multidimensional matrices.
RESULTS
1. For multidimensional matrices [2], we will consider which conditions are fulfilled for divisors and nilpotency of zero, and we will construct a nonlinear operator using those matrices. Of course, for zero to have divisors, the following condition must be fulfilled A • B = 0 . We calculate from the property of multidimensional multiplication.
A =
a,
V &6
7
a, a.
V &2
a
3
B =
b5 b8 bi b
4 y
b6 b7 V b2 b3 y
F
i=i
INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 1 ISSUE 5 UIF-2022: 8.2 | ISSN: 2181-3337
f a 1 T1 = T 2 f m A F1 = f a N F21= f c 1
=
y c y y p y y m y y p y
f b ' T 2 = T 2 f « 1 F12 = f b 1 F1 = f d 1
= y d v y q, y « y y q y
T =
t = T2
T1 • F1 T 2 • F2 "
t;1 • F21 T;2 • F22 _
T • F/ T22 • F;2'
T1 • f1 T2 • F;2
^^ + ab ab + arjb^
a5b6 + ab ab + ab
ab + ab ab + a3b4
ab + ab ab + ab3
A • B =
ab + ab ab + ab
y ab + ab ab + ab y
ab + a2bx ab + a3b4 ab + ab ab + ab
3 y
ab + ab = 0, ab + ^b = 0, ab + a2b = 0, ab + ab = 0 ab + aèb = 0, ab + ab = 0, ab + ab = 0, ab + ab = 0
When we solve this system of equations, the possible solutions are as follows.
0,0,0,0,0,0,0,0, b, b, b, b, b, b, b, b 0,0, c, d ,0,0, a, a, b, b ,0,0, b, b ,0,0 a, a ,0,0, a, a ,0,0,0,0, b, b4 ,0,0, b, b
0,0,0,0,0,0,0,0
2. In our next problem, we will consider nilpotency. « = 2, A2 = 0
F
fa b 1 f m « 1 f a b 1 f c d 1
A = y c d y A =
y p q y T y m « y y p q,
T11 =
T12 =
y c y
y d y
T1 = T 2
T2 = T 2
m
y P f «
y q y
F1 =
F;2 =
y m y
y « y
F1 = 1 2
F1 =
y p y
f d 1
y q y
T
T = T 2
T1 • F1 T2 • F2
Ti 11 Ti 11
T1 • F1 T2 • F2
Ti 1 2 Ti 1 2 _
T1 • F1 T2 • F2
T 2 1 1 T 2 1 1
T1 • F1 T2 • F2
T2 1 2 T2 1 2
A2
/ j 9 \
a + mc b + d« ac + pc bd + dq
a2 + mc b2 + d« ac + pc bd + dq
ma + mp nb + qn mc + p2 «d + q2
^ma + mp nb + q«^
2 „J , „2
mc + p «d + q2
y
2
INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 1 ISSUE 5 UIF-2022: 8.2 | ISSN: 2181-3337
a2 + mc = 0, ma + mp = 0
ac + pc = 0, mc + p2 = 0
b2 + dn = 0, nb + qn = 0
bd + dq = 0, nd + q2 = 0 3. The solution can be in the following forms. (0,0,0,0,0,0,0,0), (0,0,0,0,0, n,0,0), (0,0,0,0, m,0,0,0), (0,0,0,0, m, n,0,0),
b2 b2 a2
(0, b,0,--,0, n,0, -b), (0, b,0,--, m, n,0, -b), (a,0,--,0, m,0, -a,0),
n n m
a2 a2 b2
(a,0,--,0, m, n, -a,0), (a, b,--,--, m, n, -a, -b).
m m n
This is our problem, when does the multiplication of the matrix by itself become equal to
itself again A2 = A ?
A =
A2 =
^ a b^ v c d j
K p q
A =
'a b^
v m n j
i c d)
kP qy
a2 + mc b2 + dn ac + pc bd + dq
rma + mp nb + qn^ mc + p2 nd + q2
/
2
a + mc = a, ma + mp = m
ac + pc = c, mc + p2 = p
b2 + dn = b, nb + qn = n
bd + dq = d, nd + q2 = q (a, b, c, d, m, n, p, q) = {(1,0,0,0,0,0,0,0),(1,0,0,0,0,0,0,1), (1,0,0,0,0,0,1,0), (1,0,0,0,0,0,1,1), (1,0,0,0,0, n ,0,1), (1,0,0,0,0, n ,1,1), (1,0,0,0, m,0,0,0), (1,0,0,0, m, 0,0,1), (1,0,0,0, m, n,0,1),
(1,1,0,0,0,0,1,1), (1,1,0,0,0, n ,0,0), (1,1,0,0,0, n ,1,0),
(1,1,0,0, m ,0,0,0),(1,1,0,0, m ,0,0,1),(0,0,0,0,0,0,0,0),
(0,0,0,0,0,0,0,1), (0,0,0,0,0,0,1,0), (0,0,0,0,0,0,1,1), (0,0,0,0,0, n,0,1),
(0,0,0,0,0, n,1,1), (0,0,0,0, m, 0,1,0), (0,0,0,0, m,0,1,1), (0,0,0,0, m, n,1,1),
(0,1,0,0,0,0,0,0), (0,1,0,0,0,n,1,1)(0,1,0,0,m,0,1,0),(0,1,0,0,m,0,1,0),
(0,1,0,0,0,0,1,0),(0,1,0,0,0,0,1,1),(0,1,0,0,0, n ,0,0),(0,1,0,0,0, n ,0,0),(0,1,0,0,0, n,1,0),
(0,1,0,0, m, 0,1,1), (0,1,0,0, m, n, 1,0), (0, b, 0,b(1 ~ b) ,0, n,0,1 - b), ,
n
(0, b, 0, b(1-b), m, n,1,1 - b), (1,1,0,0, m, n,0,0), (1, b, 0, b(1-b) ,0, n,0,1 -b):
n
n
INTERNATIONAL SCIENTIFIC JOURNAL VOLUME 1 ISSUE 5 UIF-2022: 8.2 | ISSN: 2181-3337
(1, b,0,b) ,0, «,1,1 - b),(1, b,0,b), m, n,0,1 - b),(a,0,,0, m,0,1 - a,0), n n m
(a,0,a(1 - a) ,0, m,0,1 - a,1),(a,0,a(1 - a) ,0, m, n,1 - a,1),(a,1,a(1 - a) ,0, m,0,1 - a,0), m m m
(0, b,0, ,0, «,1,1 - b),(a,1,a(1 - a) ,0, m,0,1 - a,1),(a,1,a(1 - a) ,0, m, n,1 - a,0)
n m m
a(1 - a) b(1 - b)
(a, b,-,-, m, n,1 - a,1 - b).
m n
DISCUSSION
We have shown that there are many solutions for nilpotency and zero divisors. Through these for each matrix, a nonlinear stochastic operator [4]-[14] can be construct. Example: If coefficient (1,1,0,0,0,0,1,1) for a three-dimensional matrix, from (3) we see its expression on the
V (X ) = X\ + X1X2
operator. .
V (X ) — X2 ^ %2
CONCLUSIONS
In short, we want calculate multiplication of matrix, zero divisors, nilpotency and construction of an operator using three-dimensional matrices.
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