arXiv subjects

Haixia Chang

Publications and source records attributed to Haixia Chang.

3 recordsLinked to original sources

arXiv2019

Inverse eigenproblems and approximation problems for the generalized reflexive and antireflexive matrices with respect to a pair of generalized reflection matrices

A matrix $P$ is said to be a nontrivial generalized reflection matrix over the real quaternion algebra $\mathbb{H}$ if $P^{\ast }=P\neq I$ and $P^{2}=I$ where $\ast$ means conjugate and transpose. We say that $A\in\mathbb{H}^{n\times n}$ is generalized reflexive (or generalized antireflexive) with respect to the matrix pair $(P,Q)$ if $A=PAQ$ $($or $A=-PAQ)$ where $P$ and $Q$ are two nontrivial generalized reflection matrices of demension $n$. Let ${\large φ}$ be one of the following subsets of $\mathbb{H}^{n\times n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let $Z\in\mathbb{H}^{n\times m}$ with rank$\left( Z\right) =m$ and $Λ=$ diag$\left( λ_{1},...,λ_{m}\right) .$ The inverse eigenproblem is to find a\ matrix $A$ such that the set ${\large φ}\left( Z,Λ\right) =\left\{ A\in{\large φ}\text{ }|\text{ }AZ=ZΛ\right\} $ nonempty and find the general expression of $A.$\newline In this paper, we investigate the inverse eigenproblem ${\large φ}\left( Z,Λ\right) $. Moreover, the approximation problem: $\underset{A\in{\large φ}}{\min\left\Vert A-E\right\Vert _{F}}$ is studied, where $E$ is a given matrix over $\mathbb{H}$\ and $\parallel \cdot\parallel_{F}$ is the Frobenius norm.

math.RA↗

arXiv2019

Generalized low rank approximation to the symmetric positive semidefinite matrix

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$\underset{ rank(X)\leq k}{\min} \sum^m_{i=1}\left \Vert A_i - B_i XB_i^T \right \Vert^2_F,$$ where $X$ is an unknown symmetric positive semidefinite matrix and $k$ is a positive integer. We firstly use the property of a symmetric positive semidefinite matrix $X=YY^T$, $Y$ with order $n\times k$, to convert the generalized low rank approximation into unconstraint generalized optimization problem. Then we apply the nonlinear conjugate gradient method to solve the generalized optimization problem. We give a numerical example to illustrate the numerical algorithm is feasible.

math.OC↗

arXiv2016

Polytopes of Stochastic Tensors

Considering $n\times n\times n$ stochastic tensors $(a_{ijk})$ (i.e., nonnegative hypermatrices in which every sum over one index $i$, $j$, or $k$, is 1), we study the polytope ($Ω_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $Ω_{n}$ with some positive diagonals, and the polytope ($Δ_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $Ω_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $Ω_{n}$.

math.CO↗