探討當W(A) 與 W(A^k) 相等，對於所有 1 ≤ k ≤ n + 1。我們根據方陣A的unitary-similarity-invariant結構來尋找A的條件。
我們首先呈現當2×2矩陣A再一次到三次時W(A)皆相等，若且唯若A為冪等(idempotent)。則當3×3矩陣A在一次到四次時W(A)皆相等，若且唯若A么正相似(unitarily similar)於2×2冪等方正B與矩陣C的直和，且矩陣C滿足W(C^k) ⊆ W(B) 對於所有 1 ≤ k ≤ 4。我們的對於4×4矩陣的主結果將延續這個方向進行討論。

In this thesis, we are interested in the question of when $W(A)$ equals $W(A^k)$ for all $1\le k\le n+1$. We look for conditions in terms of the unitary-similarity-invariant structure of $A$. We show that if $A$ is $2\times 2$, then $W(A)=W(A^k)$ for all $1\le k\le 3$ if and only if $A$ is idempotent. We also show that if $A$ is $3\times 3$, then $W(A)=W(A^k)$ for all $1\le k\le 4$ if and only if $A$ is unitarily similar to a direct sum of the form $B\oplus C$, where $B$ is a $2\times 2$ idempotent and $C$ satisfies $W(C^k)\subseteq W(B)$ for all $1\le k\le 4$. Our main results are the analysis of $4 \times 4$ matrices along this line.

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