arXiv subjects

Ming-Cheng Tsai

Publications and source records attributed to Ming-Cheng Tsai.

17 recordsLinked to original sources

arXiv2025

Multiplicative trace and spectrum preservers on stochastic matrices

We characterize maps $ϕ_i: \mathcal{S} \to \mathcal{S}$, $i=1, \ldots, m$ and $m\ge 1$, that have the multiplicative spectrum or trace preserving property: \begin{eqnarray*} \textrm{spec} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{spec} (A_1\cdots A_m),\quad\text{or}\quad \textrm{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m)) &=& \textrm{tr} (A_1\cdots A_m), \end{eqnarray*} where $\mathcal{S}$ is the set of $n\times n$ doubly stochastic, row stochastic, or column stochastic matrices, or the space spanned by one of these sets. Linearity is assumed when $m=1$. We show that every stochastic matrix contains a real doubly stochastic component that carries the spectral information. In consequence, the multiplicative spectrum or trace preservers on these sets $ \mathcal{S} $ are linked to the corresponding preservers on the space of doubly stochastic matrices. Moreover, when $m\ge 3$, multiplicative trace preservers always coincide with multiplicative spectrum preservers.

math.FA↗

arXiv2024

Linear maps preserving $\ell_p$-norm parallel vectors

Two vectors $x, y$ in a normed vector space are parallel if there is a scalar $μ$ with $|μ| = 1$ such that $\|x+μy\| = \|x\| + \|y\|$; they form a triangle equality attaining (TEA) pair if $\|x+y\| = \|x\| + \|y\|$. In this paper, we characterize linear maps on $F^n=R^n$ or $C^n$, equipped with the $\ell_p$-norm for $p \in [1, \infty]$, preserving parallel pairs or preserving TEA pairs. Indeed, any linear map will preserve parallel pairs and TEA pairs when $1< p <\infty$. For the $\ell_1$-norm, TEA preservers form a semigroup of matrices in which each row has at most one nonzero entries; adding rank one matrices to this semigroup will be the semigroup of parallel preserves. For the $\ell_\infty$-norm, a nonzero TEA preserver, or a parallel preserver of rank greater than one, is always a multiple of an $\ell_\infty$-norm isometry, except when $F^n = R^2$. We also have a characterization for the exceptional case. The results are extended to linear maps of the infinite dimensional spaces $\ell_1(Λ)$, $c_0(Λ)$ and $\ell_\infty(Λ)$.

math.FA↗

arXiv2024

Linear maps on matrices preserving parallel pairs

Two (real or complex) $m\times n$ matrices $A$ and $B$ are said to be parallel (resp. triangle equality attaining, or TEA in short) with respect to the spectral norm $\|\cdot\|$ if $\|A+ μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ|=1$ (resp. $μ=1$). We study linear maps $T$ on $m\times n$ matrices preserving parallel (resp. TEA) pairs, i.e., $T(A)$ and $T(B)$ are parallel (resp. TEA) whenever $A$ and $B$ are parallel (resp. TEA). It is shown that when $m,n \ge 2$ and $(m,n) \ne (2,2)$, a nonzero linear map $T$ preserving TEA pairs if and only if it is a positive multiple of a linear isometry, namely, $T$ has the form $$(1) \quad A \mapsto γUAV \quad \quad \text{or} \quad \quad (2) \quad A \mapsto γUA^{t} V \quad (\text{in this case}, m = n),$$ for a positive number $γ$, and unitary (or real orthogonal) matrices $U$ and $V$ of appropriate sizes. Linear maps preserving parallel pairs are those carrying form (1), (2), or the form $$ (3) \ A \mapsto f(A) Z$$ for a linear functional $f$ and a fixed matrix $Z$. The case when $(m,n) = (2,2)$ is more complicated. There are linear maps of $2\times 2$ matrices preserving parallel pairs or TEA pairs neither of the form (1), (2) nor (3) above. Complete characterization of such maps is given with some intricate computation and techniques in matrix groups.

math.RA↗

arXiv2023

Angle-of-Arrival Estimation of Narrow Gaussian Beams for Mobile FSO Platforms

Due to the narrow beamwidths of laser Gaussian beams, accurate tracking of laser beam's angle-of-arrival is an important problem in mobile free-space optical communications. In most optical receivers today, fine tracking of angle-of-arrival involves estimating the location of the focused beam spot projected onto a focal plane array. However, for very thin Gaussian beams, both the location as well as the energy of the spot varies considerably with the variation of angle-of-arrival. In this study, we have analyzed the relationship between the angle-of-arrival and the energy of laser spot on the focal plane. We then exploited this relationship to enhance the angle-of-arrival estimation performance of our proposed receiver that takes into account both the location as well as the energy of the laser spot while estimating the angle-of-arrival. The derived Cramer-Rao bounds indicate that the system performance can be enhanced significantly for narrow Gaussian beams when both the spot location and energy are exploited for angle-of-arrival estimation.

eess.SP↗

arXiv2023

Nonsurjective zero product preservers between matrix spaces over an arbitrary field

A map $Φ$ between matrices is said to be zero product preserving if $$ Φ(A)Φ(B) = 0 \quad \text{whenever}\quad AB = 0. $$ In this paper, we give concrete descriptions of an additive/linear zero product preserver $Φ: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ between matrix algebras of different dimensions over an arbitrary field $\mathbb{F}$. In particular, we show that if $Φ$ is linear and preserves zero products then $$ Φ(A)= S\begin{pmatrix} R_1 \otimes A & 0 \cr 0 & Φ_0(A)\end{pmatrix} S^{-1}, $$ for some invertible matrices $R_1$ in ${\bf M}_k(\mathbb{F})$, $S$ in ${\bf M}_r(\mathbb{F})$ and a zero product preserving linear map $Φ_0: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_{r-nk}(\mathbb{F})$ into nilpotent matrices. If $Φ(I_n)$ is invertible, then $Φ_0$ is vacuous. In general, the structure of $Φ_0$ could be quite arbitrary, especially when $Φ_0({\bf M}_n(\mathbb{F}))$ has trivial multiplication, i.e., $Φ_0(X)Φ_0(Y) = 0$ for all $X, Y$ in ${\bf M}_n(\mathbb{F})$. We show that if $Φ_0(I_n) = 0$ or $r-nk \le n+1$, then $Φ_0({\bf M}_n(\mathbb{F}))$ indeed has trivial multiplication. More generally, we characterize subspaces ${\bf V}$ of square matrices satisfying $XY = 0$ for any $X, Y \in {\bf V}$. Similar results for double zero product preserving maps are obtained.

math.RA↗

arXiv2023

Linear maps preserving matrices annihilated by a fixed polynomial

Let ${\bf M}_n(\mathbb{F})$ be the algebra of $n\times n$ matrices over an arbitrary field $\mathbb{F}$. We consider linear maps $Φ: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ preserving matrices annihilated by a fixed polynomial $f(x) = (x-a_1)\cdots (x-a_m)$ with $m\ge 2$ distinct zeroes $a_1, a_2, \ldots, a_m \in \mathbb{F}$; namely, $$ f(Φ(A)) = 0\quad\text{whenever} \quad f(A) = 0. $$ Suppose that $f(0)=0$, and the zero set $Z(f) =\{a_1, \dots, a_m\}$ is not an additive group. Then $Φ$ assumes the form \begin{align}\label{eq:standard} A \mapsto S\begin{pmatrix} A \otimes D_1 &&\cr & A^{T} \otimes D_2& \cr && 0_s\cr\end{pmatrix}S^{-1}, \tag{$\dagger$} \end{align} for some invertible matrix $S\in {\bf M}_r(\mathbb{F})$, invertible diagonal matrices $D_1\in {\bf M}_p(\mathbb{F})$ and $D_2\in {\bf M}_q(\mathbb{F})$, where $s=r-np-nq\geq 0$. The diagonal entries $λ$ in $D_1$ and $D_2$, as well as $0$ in the zero matrix $0_s$, are zero multipliers of $f(x)$ in the sense that $λZ(f) \subseteq Z(f)$. In general, assume that $Z(f) - a_1$ is not an additive group. If $Φ(I_n)$ commutes with $Φ(A)$ for all $A\in {\bf M}_n(\mathbb{F})$, or if $f(x)$ has a unique zero multiplier $λ=1$, then $Φ$ assumes the form \eqref{eq:standard}. The above assertions follow from the special case when $f(x) = x(x-1)=x^2-x$, for which the problem reduces to the study of linear idempotent preservers. It is shown that a linear map $Φ: {\bf M}_n(\mathbb{F}) \rightarrow {\bf M}_r(\mathbb{F})$ sending disjoint rank one idempotents to disjoint idempotents always assume the above form \eqref{eq:standard} with $D_1=I_p$ and $D_2=I_q$, unless ${\bf M}_n(\mathbb{F}) = {\bf M}_2(\mathbb{Z}_2)$.

math.FA↗

arXiv2022

Linear k-power preservers and trace of power-product preservers

Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A\in V$, $B\in S$, and $ψ$ is linear" or "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A, B\in S$ and both $ϕ$ and $ψ$ are linear." The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.

math.FA↗

arXiv2022

Maps preserving trace of products of matrices

We prove the linearity and injectivity of two maps $ϕ_1$ and $ϕ_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(ϕ_1(A)ϕ_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $ϕ_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.

math.FA↗

arXiv2020

On triangular similarity of nilpotent triangular matrices

Let $B_n$ (resp. $U_n$, $N_n$) be the set of $n\times n$ nonsingular (resp. unit, nilpotent) upper triangular matrices. We use a novel approach to explore the $B_n$-similarity orbits in $N_n$. The Belitski\uı's canonical form of $A\in N_n$ under $B_n$-similarity is in $QU_n$ where $Q$ is the subpermutation such that $A\in B_n QB_n$. Using graph representations and $U_n$-similarity actions stablizing $QU_n$, we obtain new properties of the Belitski\uı's canonical forms and present an efficient algorithm to find the Belitski\uı's canonical forms in $N_n$. As consequences, we construct new Belitski\uı's canonical forms in all $N_n$'s, list all Belitski\uı's canonical forms for $n=7, 8$, and show examples of 3-nilpotent Belitski\uı's canonical forms in $N_n$ with arbitrary numbers of parameters up to $\operatorname{O}(n^2)$.

math.RT↗

arXiv2019

Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $Φ: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $Φ(A), Φ(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $Φ$ preserves disjointness if and only if $Φ$ is of the form $$Φ(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$ for some unitary matrices $U \in M_{r,r}$ and $V\in M_{s,s}$, and positive diagonal matrices $Q_1, Q_2$, where $Q_1$ or $Q_2$ may be vacuous. The result is used to characterize nonsurjective linear maps that preserve the $JB^*$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\} = \frac{1}{2}(AB^*C+CB^*A)$. The result is also applied to characterize linear maps between rectangular matrix spaces of different sizes preserving the Schatten $p$-norms or the Ky Fan $k$-norms.

math.RA↗

arXiv2019

Distributed User Clustering and Resource Allocation for Imperfect NOMA in Heterogeneous Networks

In this paper, we propose a distributed cluster formation (CF) and resource allocation (RA) framework for non-ideal non-orthogonal multiple access (NOMA) schemes in heterogeneous networks. The imperfection of the underlying NOMA scheme is due to the receiver sensitivity and interference residue from non-ideal successive interference cancellation (SIC), which is generally characterized by a fractional error factor (FEF). Our analytical findings first show that several factors have a significant impact on the achievable NOMA gain. Then, we investigate fundamental limits on NOMA cluster size as a function of FEF levels, cluster bandwidth, and quality of service (QoS) demands of user equipments (UEs). Thereafter, a clustering algorithm is developed by taking feasible cluster size and channel gain disparity of UEs into account. Finally, we develop a distributed alpha-fair RA framework where alpha governs the trade-off between maximum throughput and proportional fairness objectives. Based on the derived closed-form optimal power levels, the proposed distributed solution iteratively updates bandwidths, clusters, and UEs' transmission powers. Numerical results demonstrate that proposed solutions deliver a higher spectral and energy efficiency than traditionally adopted basic NOMA cluster size of two. We also show that an imperfect NOMA cannot always provide better performance than orthogonal multiple access under certain conditions. Finally, our numerical investigations reveal that NOMA gain is maximized under downlink/uplink decoupled (DUDe) UE association.

cs.NI↗

arXiv2018

Distributed Cluster Formation and Power-Bandwidth Allocation for Imperfect NOMA in DL-HetNets

In this paper, we consider an non-ideal successive interference cancellation (SIC) receiver based imperfect non-orthogonal multiple access (NOMA) schemes whose performance is limited by three factors: 1) Power disparity \& sensitivity constraints (PDSCs), 2) Intra-cluster interference (ICRI), and 3) Intercell-interference (ICI). By quantifying the residual interference with a fractional error factor (FEF), we show that NOMA cannot always perform better than orthogonal multiple access (OMA) especially under certain receiver sensitivity and FEF levels. Assuming the existence of an offline/online ICI management scheme, the proposed solution accounts for the ICI which is shown to deteriorate the NOMA performance particularly when it becomes significant compared to the ICRI. Then, a distributed cluster formation (CF) and power-bandwidth allocation (PBA) approach are proposed for downlink (DL) heterogeneous networks (HetNets) operating on the imperfect NOMA. We develop a hierarchically distributed solution methodology where BSs independently form clusters and distributively determine the power-bandwidth allowance of each cluster. A generic CF scheme is obtained by creating a multi-partite graph (MPG) via partitioning user equipments (UEs) with respect to their channel gains since NOMA performance is primarily determined by the channel gain disparity of cluster members. A sequential weighted bi-partite matching method is proposed for solving the resulted weighted multi-partite matching problem. Thereafter, we present a hierarchically distributed PBA approach which consists of the primary master, secondary masters, and slave problems...

cs.NI↗

arXiv2016

Trace and determinant preserving maps of matrices

Suppose a map $ϕ$ on the set of positive definite matrices satisfies $\det(A+B)=\det(ϕ(A)+ϕ(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(ϕ(A){ϕ(B)}^{-1}).$$ Through this viewpoint, we show that $ϕ$ is of the form $ϕ(A)= M^*AM$ or $ϕ(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $ϕ: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.

math.RA↗

arXiv2015

Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary

In their 2008 paper Gau and Wu conjectured that the numerical range of a 4-by-4 nilpotent matrix has at most two flat portions on its boundary. We prove this conjecture, establishing along the way some additional facts of independent interest. In particular, a full description of the case in which these two portions indeed materialize and are parallel to each other is included.

math.FA↗

arXiv2014

The spectrum of the product of operators, and the product of their numerical ranges

We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ σ(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to show that the equivalence conditions may fail in general. We then obtain conditions to identify other classes of operators $A$ so that equivalence conditions hold.

math.FA↗

arXiv2014

Factoring a quadratic operator as a product of two positive contractions

Let $T$ be a quadratic operator on a complex Hilbert space $H$. We show that $T$ can be written as a product of two positive contractions if and only if $T$ is of the form $$aI \oplus bI \oplus\begin{pmatrix} aI & P \cr 0 & bI \cr \end{pmatrix} \quad \text{on} \quad H_1\oplus H_2\oplus (H_3\oplus H_3)$$ for some $a, b\in [0,1]$ and strictly positive operator $P$ with $\|P\| \le |\sqrt{a} - \sqrt{b}|\sqrt{(1-a)(1-b)}.$ Also, we give a necessary condition for a bounded linear operator $T$ with operator matrix $\begin{pmatrix} T_1 & T_3\\ 0 & T_2\cr\end{pmatrix}$ on $H\oplus K$ that can be written as a product of two positive contractions.

math.FA↗

arXiv2012

Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form $$[{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}],$$ where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that is, for some fixed $k$, $1\le k \le \lfloor n/2\rfloor$, $n$ is divisible by $k$, and $|a_j|=|a_{k+j}|$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1...a_n=b_1...b_n$ and $S_r(|a_1|^2,..., |a_n|^2)=S_r(|b_1|^2,..., |b_n|^2)$ for all $1\le r\le \lfloor n/2\rfloor$, where $S_r$'s are the circularly symmetric functions.

math.FA↗