The Linear Discriminant Analysis (LDA) technique is an important and well-developed area of classification, and to date many linear (and also nonlinear) discrimination methods have been put forward. A complication in applying LDA to real data occurs when the number of features exceeds that of observations. In this case, the covariance estimates do not have full rank, and thus cannot be inverted. There are a number of ways to deal with this problem. In this paper, we propose improving LDA in this...

The set of all $m\times n$ Boolean matrices is denoted by ${𝕄}_{m,n}$. We call a matrix $A\in {𝕄}_{m,n}$ regular if there is a matrix $G\in {𝕄}_{n,m}$ such that $AGA=A$. In this paper, we study the problem of characterizing linear operators on ${𝕄}_{m,n}$ that strongly preserve regular matrices. Consequently, we obtain that if $min\{m,n\}\le 2$, then all operators on ${𝕄}_{m,n}$ strongly preserve regular matrices, and if $min\{m,n\}\ge 3$, then an operator $T$ on ${𝕄}_{m,n}$ strongly preserves regular matrices if and only if there are invertible matrices $U$ and $V$ such that $T\left(X\right)=UXV$ for all $X\in {𝕄}_{m,n}$, or $m=n$ and $T\left(X\right)=U{X}^{T}V$ for all $X\in {𝕄}_n$.

We introduce linear transformations of Euclidean topological spaces given by a transformation matrix. Next, we prove selected properties and basic arithmetic operations on these linear transformations. Finally, we show that a linear transformation given by an invertible matrix is a homeomorphism.

We prove a number of theorems concerning various notions used in the theory of continuity of barycentric coordinates.