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 $ntimes k$, 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.

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 $P^{ast }=P eq I$ and $P^{2}=I$ where $ast$ means conjugate and transpose. We say that $Ainmathbb{H}^{ntimes n}$ 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 ${large varphi}$ be one of the following subsets of $mathbb{H}^{ntimes n}$ : (i) generalized reflexive matrix; (ii)reflexive matrix; (iii) generalized antireflexive matrix; (iiii) antireflexive matrix. Let $Zinmathbb{H}^{ntimes m}$ with rank$left( Zright) =m$ and $Lambda=$ diag$left( lambda_{1},...,lambda_{m}right) .$ The inverse eigenproblem is to find a matrix $A$ such that the set ${large varphi }left( Z,Lambdaright) =left{ Ain{large varphi}text{ }|text{ }AZ=ZLambdaright} $ nonempty and find the general expression of $A.$ ewline In this paper, we investigate the inverse eigenproblem ${large varphi}left( Z,Lambdaright) $. Moreover, the approximation problem: $underset{Ain{large varphi}}{minleftVert A-ErightVert _{F}}$ is studied, where $E$ is a given matrix over $mathbb{H}$ and $parallel cdotparallel_{F}$ is the Frobenius norm.

Polytopes of Stochastic Tensors

Considering $ntimes ntimes n$ 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 ($Omega_{n}$) of all these tensors, the convex set ($L_n$) of all tensors in $Omega_{n}$ with some positive diagonals, and the polytope ($Delta_n$) generated by the permutation tensors. We show that $L_n$ is almost the same as $Omega_{n}$ except for some boundary points. We also present an upper bound for the number of vertices of $Omega_{n}$.