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We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$. We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.
By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].
Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$, we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $, and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $. We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$. This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc.79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.
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.
The purpose of this paper is twofold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua–Marcus’ inequalities. Our results are stronger and more general than the existing ones.
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.
A tight frame is the orthogonal projection of some orthonormal basis of $\mathbb {R}^n$ onto $\mathbb {R}^k.$ We show that a set of vectors is a tight frame if and only if the set of all cross products of these vectors is a tight frame. We reformulate a range of problems on the volume of projections (or sections) of regular polytopes in terms of tight frames and write a first-order necessary condition for local extrema of these problems. As applications, we prove new results for the problem of maximization of the volume of zonotopes.
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
$k+1$
.