In this paper, I acquaint with the thought of Centro-Symmetric using Fuzzy matrices (FM). Also I discuss the characterizations of symmetric, skew-symmetric, centro-symmetric and skew centro-symmetric ...Fuzzy matrices. Further I discuss some properties of the compact form of K-Centro symmetric matrices.
It is worth noting that the traditional methods for performing both the dimensional reduction and the classification are via the two steps iterative approaches. In this case, performing the ...dimensional reduction does not consider the classification. On the other hand, the classification is performed in the original feature domain and it does not consider the dimensional reduction. Here, the transform matrix only takes an effect on the dimensional reduction, but not on the classification. The synergy between the dimensional reduction and the classification has been ignored. As a result, the overall performance is not optimal. To address this issue, this paper proposes a joint principal component analysis (PCA) and supervised k means approach for performing the dimensional reduction and the classification simultaneously. In particular, both the reconstruction error due to the dimensional reduction as well as the total distance between the cluster centers and the feature vectors in the transformed domain are minimized subject to the unitary condition of the transform matrix. Here, we have two decision variables. They are the transform matrix and the cluster centers, instead of a single decision variable in each iteration in the traditional iterative method. To find the analytical solution of the optimization problem, the first order derivative condition of the optimization problem is first expressed as the matrix equations. However, there is a structure deficiency on the matrix equation. To address this issue, this paper employs the property of the singular matrices of the symmetric matrix for solving these matrix equations with the guarantee of the satisfaction of the structural deficiency. As a result, the analytical form of the solutions is derived. The proposed method is evaluated via performing the mental arithmetic classification based on the electroencephalograms (EEGs) downloaded from the PhysioNet database. The comparisons to the state of the art algorithms for performing the mental arithmetic classification and the conventional methods for finding the solutions of the constrained optimization problems are conducted. The results demonstrate that our proposed method achieves the higher accuracy and requires the lower execution time. This validates the effectiveness and the efficiency of our proposed method.
The security of images is one of the predominant pivotal aspects in the mammoth and still expanding digital domain. Due to chaotic system properties i.e. randomness and unpredictability is very ...appropriate to encrypt the images. In this research article, we construct an encryption model via 6D hyperchaotic map and a symmetric matrix for both color and grayscale images. We utilize the 6D hyperchaotic map in the confusion stage to change the pixel location and the symmetric matrix is used for changing the pixel value in the diffusion step for each RGB channel extraction from plain or original image. The image encryption model is checked over differential attacks (NPCR and UACI). Histogram analysis, correlation coefficients, and entropy analysis are also performed as statistical attacks. In conclusion, the image pixels are uniformly distributed, and the average entropy value are 7.9992 and 7.9973 for color and grayscale images, subsequently. The average NPCR and UACI for color images are 99.5956 and 33.4061, correspondingly, while the values for grayscale images are 99.5934 and 33.3054, respectively. These values are in the vicinity of optimal ranges. The suggested scheme's great efficiency and the proposed algorithm's resilience to a wide range of cryptanalytic attacks are implied by experimental results, statistical analysis, and differential attacks.
Let R∈Cm×m and S∈Cn×n be nontrivial k-involutions if their minimal polynomials are both xk−1 for some k≥2, i.e., Rk−1=R−1≠±I and Sk−1=S−1≠±I. We say that A∈Cm×n is (R,S,μ)-symmetric if RAS−1=ζμA, and ...A is (R,S,α,μ)-symmetric if RAS−α=ζμA with α,μ∈{0,1,…,k−1} and α≠0. Let S be one of the subsets of all (R,S,μ)-symmetric and (R,S,α,μ)-symmetric matrices. Given X∈Cn×r, Y∈Cs×m, B∈Cm×r and D∈Cs×n, we characterize the matrices A in S that minimize ‖AX−B‖2+‖YA−D‖2 (Frobenius norm) under the assumption that R and S are unitary. Moreover, among the set S(X,Y,B,D)⊂S of the minimizers of ‖AX−B‖2+‖YA−D‖2=min, we find the optimal approximate matrix A∈S(X,Y,B,D) that minimizes ‖A−G‖ to a given unstructural matrix G∈Cm×n. We also present the necessary and sufficient conditions such that AX=B,YA=D is consistent in S. If the conditions are satisfied, we characterize the consistent solution set of all such A. Finally, a numerical algorithm and some numerical examples are given to illustrate the proposed results.
Denote by
Σ
n
and
Q
n
the set of all
n
×
n
symmetric and skew-symmetric matrices over a field
F
, respectively, where
char
(
F
)
≠
2
and
|
F
|
≥
n
2
+
1
. A characterization of
ϕ
,
ψ
:
Σ
n
→
Σ
n
, ...for which at least one of them is surjective, satisfying
det
(
ϕ
(
x
)
+
ψ
(
y
)
)
=
det
(
x
+
y
)
(
x
,
y
∈
Σ
n
)
is given. Furthermore, if
n
is even and
ϕ
,
ψ
:
Q
n
→
Q
n
, for which
ψ
is surjective and
ψ
(0) = 0, satisfy
det
(
ϕ
(
x
)
+
ψ
(
y
)
)
=
det
(
x
+
y
)
(
x
,
y
∈
Q
n
)
,
then
ϕ
=
ψ
and
ψ
must be a bijective linear map preserving the determinant.
•In this paper, we proposed an algorithm to solve mixed solutions of the matrix Equation ∑i=1tAiXiBi=E with sub-matrix constraints. We also prove that the iterative solution sequence generated by the ...algorithm is convergent. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case, for both small-scale and large-scale matrices. The work belongs to the field of numerical algebra, and has been widely concerned.
We put forward and analyze in details an iterative method to find the mixed solutions of a matrix equation with sub-matrix constraints. The convergence of the approximated solution sequence generated by the iterative method is investigated, showing that if the constrained matrix equation is consistent, the mixed solution group can be obtained after a finite number of iterations. Moreover, for a given matrix, its best approximation is obtained, which is the mixed solution of the matrix equation with sub-matrix constraints. Finally, a large number of numerical experiments are carried out, and results show that the algorithm is effective not only in image restoration, but also in the general case for both small-scale and large-scale matrices.
The left and right inverse eigenvalue problem, which mainly arises in perturbation analysis of matrix eigenvalue and recursive matters, has some practical applications in engineer and scientific ...computation fields. In this paper, we give the solvability conditions of and the general expressions to the left and right inverse eigenvalue problem for the (R,S)-symmetric and (R,S)-skew symmetric solutions. The corresponding best approximation problems for the left and right inverse eigenvalue problem are also solved. That is, given an arbitrary complex n-by-n matrix A∼, find a (R,S)-symmetric (or (R,S)-skew symmetric) matrix AA∼ which is the solution to the left and right inverse eigenvalue problem such that the distance between A∼ and AA∼ is minimized in the Frobenius norm. We give an explicit solution to the best approximation problem in the (R,S)-symmetric and (R,S)-skew symmetric solution sets of the left and right inverse eigenvalue problem under the assumption that R=R∗ and S=S∗. A numerical example is given to illustrate the effectiveness of our method.
For each pair of complex symmetric matrices (A,B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A˜,B˜), close to (A,B) ...can be reduced by congruence transformation that smoothly depends on the entries of A˜ and B˜. Such a normal form is called a miniversal deformation of (A,B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A,B) to its miniversal deformation.
PDF
