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Necessary and sufficient conditions for the equality of the determinant and permanent for all powers of a given matrix are provided. The characterisation is based on a condition on merely one power.
where \[{\text{c}}(k,X)\] stands for the best constant \[C > 0\] such that \[\mathop P\limits^ \vee \leqslant CP\] for every k-homogeneous polynomial \[P \in \mathcal{P}{(^k}X)\]. We show that if X is a finite dimensional complex space then \[{\text{c}}(X) = 1\]. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.
The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak’s complexification procedure.
We also study some other properties connected with polarization. Namely, we provide necessary conditions related to the geometry of X for \[c(2,X) = 1\] to hold. Additionally we link polarization constants with certain estimates of the nuclear norm of the product of polynomials.
Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$. This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$. Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$. Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.
Associated with every $2n\times 2n$ real positive definite matrix $A,$ there exist n positive numbers called the symplectic eigenvalues of $A,$ and a basis of $\mathbb {R}^{2n}$ called the symplectic eigenbasis of A corresponding to these numbers. In this paper, we discuss differentiability and analyticity of the symplectic eigenvalues and corresponding symplectic eigenbasis and compute their derivatives. We then derive an analogue of Lidskii’s theorem for symplectic eigenvalues as an application.
We introduce the
$\textbf{h}$
-minimum spanning length of a family
${\mathcal A}$
of
$n\times n$
matrices over a field
$\mathbb F$
, where
$\textbf{h}$
is a p-tuple of positive integers, each no more than n. For an algebraically closed field
$\mathbb F$
, Burnside’s theorem on irreducibility is essentially that the
$(n,n,\ldots ,n)$
-minimum spanning length of
${\mathcal A}$
exists if
${\mathcal A}$
is irreducible. We show that the
$\textbf{h}$
-minimum spanning length of
${\mathcal A}$
exists for every
$\textbf{h}=(h_1,h_2,\ldots , h_p)$
if
${\mathcal A}$
is an irreducible family of invertible matrices with at least three elements. The
$(1,1, \ldots ,1)$
-minimum spanning length is at most
$4n\log _{2} 2n+8n-3$
. Several examples are given, including one giving a complete calculation of the
$(p,q)$
-minimum spanning length of the ordered pair
$(J^*,J)$
, where J is the Jordan matrix.
We provide a generalised Laplace expansion for the permanent function and, as a consequence, we re-prove a multinomial Vandermonde convolution. Some combinatorial identities are derived by applying special matrices to the expansion.
We study the
$L^{q}$
-spectrum of measures in the plane generated by certain nonlinear maps. In particular, we consider attractors of iterated function systems consisting of maps whose components are
$C^{1+\alpha }$
and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the
$L^{q}$
-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.
We introduce the notion of the slot length of a family of matrices over an arbitrary field
${\mathbb {F}}$
. Using this definition it is shown that, if
$n\ge 5$
and A and B are
$n\times n$
complex matrices with A unicellular and the pair
$\{A,B\}$
irreducible, the slot length s of
$\{A,B\}$
satisfies
$2\le s\le n-1$
, where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of
$n\times n$
matrices, where
$n\ge 5$
, is at most
$n-1$
. The slot length of a family of rank-one complex matrices can be equal to n.
This work focuses on the ongoing research of lineability (the search for large linear structures within certain non-linear sets) in non-Archimedean frameworks. Among several other results, we show that there exist large linear structures inside each of the following sets: (i) functions with a fixed closed subset of continuity, (ii) all continuous functions that are not Darboux continuous (or vice versa), (iii) all functions whose Dieudonné integral does not behave as an antiderivative, and (iv) functions with finite range and having antiderivative.
If A is a real
$2n \times 2n$
positive definite matrix, then there exists a symplectic matrix M such that
$M^TAM=\text {diag}(D, D),$
where D is a positive diagonal matrix with diagonal entries
$d_1(A)\leqslant \cdots \leqslant d_n(A).$
We prove a maxmin principle for
$d_k(A)$
akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality
$d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$
We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,
We investigate the real space H of Hermitian matrices in
$M_n(\mathbb{C})$
with respect to norms on
$\mathbb{C}^n$
. For absolute norms, the general form of Hermitian matrices was essentially established by Schneider and Turner [Schneider and Turner, Linear and Multilinear Algebra (1973), 9–31]. Here, we offer a much shorter proof. For non-absolute norms, we begin an investigation of H by means of a series of examples, with particular reference to dimension and commutativity.
Building on MacDonald’s formula for the distance from a rank-one projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$, we prove that the distance from a rank $n-1$ projection to the set of nilpotents in $\mathbb {M}_n(\mathbb {C})$ is $\frac {1}{2}\sec (\frac {\pi }{\frac {n}{n-1}+2} )$. For each $n\geq 2$, we construct examples of pairs $(Q,T)$ where Q is a projection of rank $n-1$ and $T\in \mathbb {M}_n(\mathbb {C})$ is a nilpotent of minimal distance to Q. Furthermore, we prove that any two such pairs are unitarily equivalent. We end by discussing possible extensions of these results in the case of projections of intermediate ranks.
We show that an irreducible family
${\mathcal{S}}$
of complex
$n\times n$
matrices satisfies Paz’s conjecture if it contains a rank-one matrix. We next investigate properties of families of rank-one matrices. If
${\mathcal{R}}$
is a linearly independent, irreducible family of rank-one matrices then (i)
${\mathcal{R}}$
has length at most
$n$
, (ii) if all pairwise products are nonzero,
${\mathcal{R}}$
has length 1 or 2, (iii) if
${\mathcal{R}}$
consists of elementary matrices, its minimum spanning length
$M$
is the smallest integer
$M$
such that every elementary matrix belongs to the set of words in
${\mathcal{R}}$
of length at most
$M$
. Finally, for any integer
$k$
dividing
$n-1$
, there is an irreducible family of elementary matrices with length
$k+1$
.
We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach
$\ast$
-algebra, analytic convex functions and positive normalised linear functionals.
Inequalities on partial traces of positive semidefinite matrices are studied. Extensions of several existing inequalities on the determinant of partial traces are then obtained. Particularly, we improve a determinantal inequality given by Lin [Canad. Math. Bull. 59(2016)].
The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$. This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$.
We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.
We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$-algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$-Drazin invertible elements of a $C^{\ast }$-algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.