一$n$階方陣$A$，存在一單位向量$x$使得對所有$k\geq 1$皆滿足 $\|A^kx\|=\|A^k\|=\|A\|$。我們刻畫了符合上述性質的矩陣，其充分必要條件為：矩陣$A$酉相似一2階方塊矩陣$\begin{bmatrix}
B& D\\
0& C
\end{bmatrix}$，其中$B$為符合$B^{*}B=B^{*^{2}}B^{2}$的$r$階方陣，且對所有$k=1,2,\ldots$都有$\| A^k \|= \| B^k \|$。除此之外，我們也刻畫了滿足$A^*A=A^{*^2}A^2$的$n$階方陣$A$。文末，我們給了一個吳培元教授、高華隆教授和王國仲副教授所猜測結果的反例。

For an $n$-by-$n$ matrix $A$, there exists a unit vector $x$ satisfying $\|A^kx\|=\|A^k\|=\|A\|$ for all $k\ge 1$.We give a characterization for such a matrix. A necessary and sufficient for this is unitarily similar to a $2$-by-$2$ block matrix $\begin{bmatrix}
B& D\\
0& C
\end{bmatrix}$, where $B\in M_r$ with $B^{*}B=B^{*^{2}}B^{2}$, and $\| A^k \|= \| B^k \|$ for $k=1,2,\ldots$. Furthermore, we also characterize $n$-by-$n$ matrices which are $A^*A=A^{*^2}A^2$. Finally, we give a counterexample to a conjecture of Gau. et al..

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