正泛函与Hessenberg矩阵

构造所有实Hessenberg酉矩阵的一个简单算法

A Simple Algorithm for Constructing all Real Hessenberg Unitary Matrices

次对角线之下为零的酉矩阵（Hessenberg酉矩阵）在分析中有许多用途。该算法给出了一组关于酉矩阵的必要条件，从而得到稀疏的酉矩阵。本文给出了构造所有实Hessenberg酉矩阵的一个算法。由该算法给出的$n\times n$酉矩阵具有$n-1$变量，可以选择这些变量以给出特定应用所需的附加性质。

Positive bidiagonal factorization of tetradiagonal Hessenberg matrices

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. In this paper conditions, in terms of continued fractions, for an oscillatory tetradiagonal Hessenberg matrix to have such positive bidiagonal factorization are found. Oscillatory tetradiagonal Toeplitz matrices are taken as a case study of matrix that admits a positive bidiagonal factorization. Moreover, it is proved that oscillatory banded Hessenberg matrices are organized in rays, with the origin of the ray not having the positive bidiagonal factoriza…

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and random walks

A spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization is found. The large knowledge on the spectral and factorization properties of oscillatory matrices leads to this spectral Favard theorem in terms of sequences of multiple orthogonal polynomials of types I and II with respect to a set of positive Lebesgue-Stieltjes~measures. Also a multiple Gauss quadrature is proven and corresponding degrees of precision are found. This spectral Favard theorem is applied to Markov chains with $(p+2)$-diagonal transition matrices, i.e. beyond bi…

Kippenhahn Varieties and the Weyl Calculus for Several Matrices I

The paper reviews properties of the Weyl functional calculus for several operators and its relation to the generalised numerical range of $n$ hermitian matrices. The support and singular support of the Weyl functional calculus for $n$ hermitian matrices are determined by Kippenhahn varieties in algebraic geometry.

关于$A\mapsto\Tr(Pf(A))$型迹泛函的方向导数

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

给定函数$f:(0,\infty)\rightarrow\RR$和半正定$n\times n$矩阵$P$，可以将正定$n\times n$矩阵上的迹泛函定义为$A\mapsto \Tr(Pf(A))$。对于可微函数$f$，函数$A\mapsto \Tr(Pf(A))$在所有正定矩阵$A$处可微。在~ $f$上一定的连续性条件下，该函数可推广到某些非正定矩阵$A$，并可在那里计算$\Tr(Pf(A)$的\emph{directional}导数.本文给出了这些方向导数存在的条件，并计算了它们。这些条件适用于函数$f(x)=\log(x)$和函数$f_p(x)=x^p$，适用于所有$p>-1$。相应的迹泛函的导数计算在这里。

函数 calculus、正性与矩阵卷积

Functional Calculi, Positivity, and Convolution of Matrices

本文提出了一种基于矩阵卷积的函数 calculus 框架，揭示了卷积与对称群 Bruhat 秩的联系，并扩展了经典正性保持理论。