Related papers: Generalized Induced Norms

We give a canonical form of m-by-2-by-2 spatial matrices for equivalence over any field.

Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem…

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…

There is growing body of learning problems for which it is natural to organize the parameters into matrix, so as to appropriately regularize the parameters under some matrix norm (in order to impose some more sophisticated prior knowledge).…

The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in…

The theorem like Pontryagin's maximum principle for multiple integrals is proved. Unlike the usual maximum principle, the maximum should be taken not over all matrices, but only on matrices of rank one. Examples are given.

We discuss two conjectures. (I) For each x_1,...,x_n \in R (C) there exist y_1,...,y_n \in R (C) such that \forall i \in {1,...,n} |y_i| \leq 2^{2^{n-2}} \forall i \in {1,...,n} (x_i=1 \Rightarrow y_i=1) \forall i,j,k \in {1,...,n}…

In this article, we are going to search for $n\times n$ matrices $A$ and $B$ such that their generalized numerical range $$W_A(B)=\{tr(AU^*BU) \ :\ U^*U=UU^*=I\}$$ is convex. More specifically, we consider $A=\hat{A}\oplus I_k$ and…

Let $m,n$ be positive integers. For all $m\times n$ complex matrices $A, C$ and an $n\times m$ matrix $B$, we define a generalized commutator as $ABC-CBA$. We estimate the Frobenius norm of it, and finally get the inequality, which is a…

We answer two questions posed 1998 in the book 'Arnolds problems'. First, over any field k there is a representative system for the similarity classes of nxn-matrices which is a finite disjoint union of affine subspaces. And second, for n>1…

An irreducible norm closed semigroup of complex matrices is simultaneously similar to a semigroup of partial isometries if and only if (a) the norms of all nonzero members of it are uniformly bounded above and below, and (b) its idempotents…

We characterize in terms of inequalities the possible generalized singular numbers of a product AB of operators A and B having given generalized singular numbers, in an arbitrary finite von Neumann algebra. We also solve the analogous…

We compare two known methods of extending a complex, unital, commutative normed algebra so as to include solutions to sets of monic polynomials over the original algebra. (One of these is a generalisation of a construction from the thesis…

Mathematically characterizing the implicit regularization induced by gradient-based optimization is a longstanding pursuit in the theory of deep learning. A widespread hope is that a characterization based on minimization of norms may…

A $n\times n$ matrix $A$ has normal defect one if it is not normal, however can be embedded as a north-western block into a normal matrix of size $(n+1)\times (n+1)$. The latter is called a minimal normal completion of $A$. A construction…

Inequalities for norms of different versions of the geometric mean of two positive definite matrices are presented.

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping…

We prove that for all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$, such that for all $q > C_k$, and for $n = 1, 2$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers and for all $n \geq 3$ every…

An $n$-correct set $\mathcal{X}$ in the plane is a set of nodes admitting unique interpolation with bivariate polynomials of total degree at most $n$. A $k$-node line is a line passing through exactly $k$ nodes of $\mathcal{X}.$ A line can…

Let $\mathcal{A}$ and $\mathcal{B}$ be two algebras, let $\mathcal{M}$ be a $\mathcal{B}$-bimodule and let $n$ be a positive integer. A linear mapping $D_n:\mathcal{A} \rightarrow \mathcal{M}$ is called a strongly generalized derivation of…