相关论文: Principal pivot transforms: properties and applica…
Positive-definite matrices materialize as state transition matrices of linear time-invariant gradient flows, and the composition of such materializes as the state transition after successive steps where the driving potential is suitably…
This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension…
We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.
The paper explains how a unit generalized quaternion is used to represent a rotation of a vector in 3-dimensional space. We review of some algebraic properties of generalized quaternions and operations between them and then show their…
In this paper, we consider a system of homogeneous algebraic equations in complex variables and their conjugates, which arise naturally from the range criterion for separability of PPT states. We examine systematically these equations to…
Principal Components Analysis is a widely used technique for dimension reduction and characterization of variability in multivariate populations. Our interest lies in studying when and why the rotation to principal components can be used…
We resolve an algebraic version of Schoenberg's celebrated theorem [Duke Math.J., 1942] characterizing entrywise matrix transforms that preserve positive definiteness. Compared to the classical real and complex settings, we consider…
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the…
The main result of this paper is a recursive description of all decompositions \[ \Delta^+ = \Phi_1 \sqcup \Phi_2 \sqcup \dots \sqcup \Phi_k \] of the positive roots $\Delta^+$ of an arbitrary root system $\Delta$ into a disjoint union of…
Big data is transforming our world, revolutionizing operations and analytics everywhere, from financial engineering to biomedical sciences. The complexity of big data often makes dimension reduction techniques necessary before conducting…
Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of…
A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for…
We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing…
The results on the inversion of convolution operators as well as Toeplitz (and block Toeplitz) matrices in the $1$-D (one-dimensional) case are classical and have numerous applications. Last year, we considered the $2$-D case of…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
The decomposition of a matrix, as a product of factors with particular properties, is a much used tool in numerical analysis. Here we develop methods for decomposing a matrix $C$ into a product $X Y$, where the factors $X$ and $Y$ are…
In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to…
Principal Moment Analysis is a method designed for dimension reduction, analysis and visualization of high dimensional multivariate data. It generalizes Principal Component Analysis and allows for significant statistical modeling…
In the last decades the Moore-Penrose pseudoinverse has found a wide range of applications in many areas of Science and became a useful tool for physicists dealing, for instance, with optimization problems, with data analysis, with the…
We consider principal pivot transform (pivot) on graphs. We define a natural variant of this operation, called dual pivot, and show that both the kernel and the set of maximally applicable pivots of a graph are invariant under this…