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### A practical application of kernel-based fuzzy discriminant analysis

International Journal of Applied Mathematics and Computer Science

A novel method for feature extraction and recognition called Kernel Fuzzy Discriminant Analysis (KFDA) is proposed in this paper to deal with recognition problems, e.g., for images. The KFDA method is obtained by combining the advantages of fuzzy methods and a kernel trick. Based on the orthogonal-triangular decomposition of a matrix and Singular Value Decomposition (SVD), two different variants, KFDA/QR and KFDA/SVD, of KFDA are obtained. In the proposed method, the membership degree is incorporated...

### A subclass of strongly clean rings

Communications in Mathematics

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let ${J}^{#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very ${J}^{#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of ${J}^{#}$. A ring $R$ is said to be very ${J}^{#}$-clean in case every element in $R$ is very ${J}^{#}$-clean. We prove that every very ${J}^{#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

### All about the ⊥ with its applications in the linear statistical models

Open Mathematics

For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

### Factorizations for q-Pascal matrices of two variables

Special Matrices

In this second article on q-Pascal matrices, we show how the previous factorizations by the summation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows [...] We also find two different matrix products for [...]

### On a $J$-polar decomposition of a bounded operator and matrices of $J$-symmetric and $J$-skew-symmetric operators.

Banach Journal of Mathematical Analysis [electronic only]

Special Matrices

The class of sparse companion matrices was recently characterized in terms of unit Hessenberg matrices. We determine which sparse companion matrices have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix and describe how to find the permutation involved. In the process, we determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to find a Fiedler...

### Remarks on eigenvalues of infinite matrices

Colloquium Mathematicae

### Some Basic Properties of Some Special Matrices. Part III

Formalized Mathematics

This article describes definitions of subsymmetric matrix, anti-subsymmetric matrix, central symmetric matrix, symmetry circulant matrix and their basic properties.

### Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Special Matrices

Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether...

### The Smith normal form of product distance matrices

Special Matrices

Let G = (V, E) be a connected graph with 2-connected blocks H1, H2, . . . , Hr. Motivated by the exponential distance matrix, Bapat and Sivasubramanian in [4] defined its product distance matrix DG and showed that det DG only depends on det DHi for 1 ≤ i ≤ r and not on the manner in which its blocks are connected. In this work, when distances are symmetric, we generalize this result to the Smith Normal Form of DG and give an explicit formula for the invariant factors of DG.

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