Positive linear maps on normal matrices

关于所有方差-协方差矩阵集合上相对于减偏序的双单调映射

Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order

本文刻画了在半正定实矩阵锥上双向保持减偏序的所有满射形式，完整解决了该偏序保持映射的分类问题。

Linear maps on $M_n(\mathbb{R})$ preserving Schur stable matrices

An $n \times n$ matrix $A$ with real entries is said to be Schur stable if all the eigenvalues of $A$ are inside the open unit disc. We investigate the structure of linear maps on $M_n(\mathbb{R})$ that preserve the collection $\mathcal{S}$ of Schur stable matrices. We prove that if $L$ is a linear map such that $L(\mathcal{S}) \subseteq \mathcal{S}$, then $ρ(L)$ (the spectral radius of $L$) is at most $1$ and when $L(\mathcal{S}) = \mathcal{S}$, we have $ρ(L) = 1$. In the latter case, the map $L$ preserves the spectral radius function and using this, we characterize such maps on both $M_n(\ma…

The Normal Defect of Some Classes of Matrices

An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices A_{ext}. We also prove a result for a related class of n \times n matrices. Finally, we present an example of a 6 \times 6 block diagonal matrix having the property that its normal defect is strictly less th…

Limit and extended limit sets of matrices in Jordan normal form

In this note we describe the limit and the extended limit sets of every vector for a single matrix in Jordan normal form.

On the sum of normal matrices

This short paper revisits a remarkable but almost overlooked result of Djoković [Proc. Amer. Math. Soc. 27 (1971) 19-23]. A connection to a result of uSemrl is pointed out. With Djoković's result, an extension of Craig-Sakamoto theorem to $k$ ($k\ge 2$) normal matrices is presented. A comment to a recent monthly problem is also given.

On the similarity of AB and BA for normal and other matrices

It is well-known that $AB$ and $BA$ are similar when $A$ and $B$ are complex square Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if $A$ is Hermitian and $B$ is normal. Perhaps surprisingly, similarity does hold when $A$ is positive semidefinite and $B$ is normal.

A Variant of the Kaplansky Problem for Maps on Positive Matrices

We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result easily generalizes to maps on hermitian matrices.

Matrices with normal defect one

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 of all matrices with normal defect one is given. Also, a simple procedure is presented which allows one to check whether a given matrix has normal defect one, and if this is the case -- to construct all its minimal normal completions. A characterization of the generic case for each $n$ under the assumption $\rank(A^*A-AA^*)=2$ (which is necessary for $A$ to h…

On directional derivatives of trace functionals of the form $A\mapsto\Tr(Pf(A))$

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ is differentiable at all positive definite matrices $A$. Under certain continuity conditions on~$f$, this function may be extended to certain non-positive-definite matrices $A$, and the \emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This note presents conditions for these directional derivatives to exist and…

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.

Self-Dual Maps and Symmetric Bistochastic Matrices

The H-unistochastic matrices are a special class of symmetric bistochastic matrices obtained by taking the square of the absolute value of each entry of a Hermitian unitary matrix. We examine the geometric relationship of the convex hull of the n by n H-unistochastic matrices relative to the larger convex set of n by n symmetric bistochastic matrices. We show that any line segment in the convex set of the n by n symmetric bistochastic matrices which passes through the centroid of this convex set must spend at least two-thirds of its length in the convex hull of the n by n H-unistochastic matri…

Maps on positive definite matrices preserving Bregman and Jensen divergences

In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.

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.

Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $$\|A\otimes B\| = \|ϕ(A\otimes B)\| \quad \hbox{for all} A \in M_m \hbox{and} B \in M_n$$ if and only if there are unitary $U, V \in M_{mn}$ such that $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$, where $φ_s(X)$ is either the identity map $X \map…

Strong $k$-commutativity preserving maps on 2$\times$2 matrices

Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$, the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with $[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map $Φ: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range containing all rank one matrices satisfies that $[Φ(A),Φ(B)]_k = [A,B]_k $ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist a functional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a sc…

Location of Ritz values in the numerical range of normal matrices

Let $μ_1$ be a complex number in the numerical range $W(A)$ of a normal matrix $A$. In the case when no eigenvalues of $A$ lie in the interior of $W(A)$, we identify the smallest convex region containing all possible complex numbers $μ_2$ for which $\begin{bmatrix}μ_1& *\\0& μ_2\end{bmatrix}$ is a $2$-by-$2$ compression of $A$.

Characterization of preorders induced by positive maps in the set of Hermitian matrices

Uhlmann showed that there exists a positive, unital and trace-preserving map transforming a Hermitian matrix $A$ into another $B$ if and only if the vector of eigenvalues of $A$ majorizes that of $B$. In this work I characterize the existence of such a transformation when one of the conditions of unitality or trace preservation is dropped. This induces two possible preorders in the set of Hermitian matrices and I argue how this can be used to construct measures quantifying the lack of positive semidefiniteness of any given Hermitian matrix with relevant monotonicity properties. It turns out th…

On indecomposable normal matrices in spaces with indefinite scalar product

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [.,.] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in terms of specific functions of v = min{v-, v+}, where v-, (v+) is the number of negative (positive) squares of the form [x,x]. All the bounds except for one are proved to be strict.

Hill representations for *-linear matrix maps

In a paper from 1973 R.D. Hill studied linear matrix maps $\mathcal{L}:\mathbb{C}^{q \times q}\to\mathbb{C}^{n \times n}$ which map Hermitian matrices to Hermitian matrices, or equivalently, preserve adjoints, i.e., $\mathcal{L}(V^*)=\mathcal{L}(V)^*$, via representations of the form \begin{equation*} \mathcal{L}(V)=\sum_{k,l=1}^m \mathbb{H}_{kl}\, A_l V A_k^*,\quad V\in\mathbb{C}^{q \times q}, \end{equation*} for matrices $A_1,\ldots,A_m \in\mathbb{C}^{n \times q}$ and continued his study of such representations in later work, sometimes with co-authors, to completely positive matrix maps and …

Hermitian和正常矩阵的特征值稳定性

Eigenvalue stability of Hermitian and normal matrices

研究Hermitian和正常矩阵特征值映射的连续性，证明其在Sobolev空间中的非均匀连续性，并探讨其在奇异值、矩阵条件数等应用中的稳定性。