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Structures and Numerical Ranges of Power Partial Isometries

We derive a matrix model, under unitary similarity, of an $n$-by-$n$ matrix $A$ such that $A, A^2, \ldots, A^k$ ($k\ge 1$) are all partial isometries, which generalizes the known fact that if $A$ is a partial isometry, then it is unitarily similar to a matrix of the form ${\scriptsize\left[\begin{array}{cc} 0 & B 0 & C\end{array}\right]}$ with $B^*B+C^*C=I$. Using this model, we show that if $A$ has ascent $k$ and $A, A^2, \ldots, A^{k-1}$ are partial isometries, then the numerical range $W(A)$ of $A$ is a circular disc centered at the origin if and only if $A$ is unitarily similar to a direct sum of Jordan blocks whose largest size is $k$. As an application, this yields that, for any $S_n$-matrix $A$, $W(A)$ (resp., $W(A\otimes A)$) is a circular disc centered at the origin if and only if $A$ is unitarily similar to the Jordan block $J_n$. Finally, examples are given to show that the conditions that $W(A)$ and $W(A\otimes A)$ are circular discs at 0 are independent of each other for a general matrix $A$.