Related papers: Companion unit lower Hessenberg matrices
Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naive assertion that a matrix is UET if and only if it is unitarily…
We call an operator algebra A {\em reversible} if A with reversed multiplication is also an abstract operator algebra (in the modern operator space sense). This class of operator algebras is intimately related to the {\em symmetric operator…
One studies certain degenerations of the generic square matrix over a field $k$ along with its main related structures, such as the determinant of the matrix, the ideal generated by its partial derivatives, the polar map defined by these…
Orthogonal matrices which are linear combinations of permutation matrices have attracted enormous attention in quantum information and computation. In this paper, we provide a complete parametric characterization of all complex, real and…
We look at Bohemian matrices, specifically those with entries from $\{-1, 0, {+1}\}$. More, we specialize the matrices to be upper Hessenberg, with subdiagonal entries $1$. Even more, we consider Toeplitz matrices of this kind. Many…
In this paper, the familiar notion of a Henselian pair is extended to the noncommutative case. Furthermore, the problem of Henselizations is studied in the noncommutative context, and it is shown that every (not necessarily commutative)…
We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions $n \geq 3$, there always exists a…
Hessenberg varieties are a family of subvarieties of the flag variety, including the Springer fibers, the Peterson variety, and the entire flag variety itself. The seminal example arises from a problem in numerical analysis and consists for…
We study an ensemble of random matrices (the Rosenzweig-Porter model) which, in contrast to the standard Gaussian ensemble, is not invariant under changes of basis. We show that a rather complete understanding of its level correlations can…
A random matrix is likely to be well conditioned, and motivated by this well known property we employ random matrix multipliers to advance some fundamental matrix computations. This includes numerical stabilization of Gaussian elimination…
An algorithm is presented for generating successive approximations to trigonometric functions of sums of non-commuting matrices. The resulting expressions involve nested commutators of the respective matrices. The procedure is shown to…
The task of analytically diagonalizing a tridiagonal matrix can be considerably simplified when a part of the matrix is uniform. Such quasi-uniform matrices occur in several physical contexts, both classical and quantum, where…
Four different extensions of the Standard Model to non-commutative space-time are considered. They all have the structure group U_Y(1) x SU_L(2) x SU_c(3) but differ through the way Yukawa interaction is implemented. Models based on…
Rapid convergence of the shifted QR algorithm on symmetric matrices was shown more than fifty years ago. Since then, despite significant interest and its practical relevance, an understanding of the dynamics and convergence properties of…
Non-Hermitian random matrices have been utilized in such diverse fields as dissipative and stochastic processes, mesoscopic physics, nuclear physics, and neural networks. However, the only known universal level-spacing statistics is that of…
Gradient descent optimizations and backpropagation are the most common methods for training neural networks, but they are computationally expensive for real time applications, need high memory resources, and are difficult to converge for…
In the design of decentralized networked systems, it is useful to know whether a given network topology can sustain stable dynamics. We consider a basic version of this problem here: given a vector space of sparse real matrices, does it…
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace is spanned by rank-1 matrices, the matrices can be obtained…
In this paper, we introduce a reduction of a matrix to a condensed form, the upper $J$- Hessenberg form, via elementary symplectic Householder transformations, which are rank-one modification of the identity . Features of the reduction are…
We revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random. These matrices have a one-to-one correspondence with the adjacency…