Ali Godjali | Searcharxiv

arXiv2011

Thin Hessenberg Pairs and Double Vandermonde Matrices

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered pair of linear transformations $A: V \rightarrow V$ and $A^*: V \rightarrow V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg. \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH pair}). By the {\it diameter} of the pair we mean the dimension of $V$ minus one. There is an "oriented" version of a TH pair called a TH system. In this paper we investigate a connection between TH systems and double Vandermonde matrices. We give a bijection between any two of the following three sets: \cdot The set of isomorphism classes of TH systems over $\K$ of diameter $d$. \cdot The set of normalized west-south Vandermonde systems in $\Mdf$. \cdot The set of parameter arrays over $\K$ of diameter $d$. We give a bijection between any two of the following five sets: \cdot The set of affine isomorphism classes of TH systems over $\K$ of diameter $d$. \cdot The set of isomorphism classes of RTH systems over $\K$ of diameter $d$. \cdot The set of affine classes of normalized west-south Vandermonde systems in $\Mdf$. \cdot The set of normalized west-south Vandermonde matrices in $\Mdf$. \cdot The set of reduced parameter arrays over $\K$ of diameter $d$.

math.RA↗

arXiv2009

Thin Hessenberg Pairs

A square matrix is called {\it Hessenberg} whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$ denote a nonzero finite-dimensional vector space over a field $\fld$. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ which satisfy both (i), (ii) below. \begin{enumerate} \item There exists a basis for $V$ with respect to which the matrix representing $A$ is Hessenberg and the matrix representing $A^*$ is diagonal. \item There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg. \end{enumerate} \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH pair}). This is a special case of a {\it Hessenberg pair} which was introduced by the author in an earlier paper. We investigate several bases for $V$ with respect to which the matrices representing $A$ and $A^*$ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an "oriented" version of $A,A^*$ called a TH system. We classify the TH systems up to isomorphism.

math.RA↗

arXiv2008

Hessenberg Pairs of Linear Transformations

Let $\fld$ denote a field and $V$ denote a nonzero finite-dimensional vector space over $\fld$. We consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to V$ that satisfy (i)--(iii) below. Each of $A, A^*$ is diagonalizable on $V$. There exists an ordering $\lbrace V_i \rbrace_{i=0}^d$ of the eigenspaces of $A$ such that A^* V_i \subseteq V_0 + V_1 + ... + V_{i+1} \qquad \qquad (0 \leq i \leq d), where $V_{-1} = 0$, $V_{d+1}= 0$. There exists an ordering $\lbrace V^*_i \rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that A V^*_i \subseteq V^*_0 + V^*_1 + ... +V^*_{i+1} \qquad \qquad (0 \leq i \leq δ), where $V^*_{-1} = 0$, $V^*_{δ+1}= 0$. We call such a pair a {\it Hessenberg pair} on $V$. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.

math.RA↗