From 6eeb626776039409b65d54f617a86e68fd84682e Mon Sep 17 00:00:00 2001 From: Cym10x Date: Wed, 21 Jan 2026 15:45:42 +0800 Subject: [PATCH] =?UTF-8?q?M=20=E7=BC=96=E5=86=99=E5=B0=8F=E7=BB=84/?= =?UTF-8?q?=E8=AE=B2=E4=B9=89/=E7=9F=A9=E9=98=B5=E7=9B=B8=E4=BC=BC?= =?UTF-8?q?=E5=8F=98=E6=8D=A2=EF=BC=88=E8=A7=A3=E6=9E=90=E7=89=88=EF=BC=89?= =?UTF-8?q?.md?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 编写小组/讲义/矩阵相似变换（解析版）.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/编写小组/讲义/矩阵相似变换（解析版）.md b/编写小组/讲义/矩阵相似变换（解析版）.md index 8dd449f..35b6a99 100644 --- a/编写小组/讲义/矩阵相似变换（解析版）.md +++ b/编写小组/讲义/矩阵相似变换（解析版）.md @@ -492,8 +492,9 @@ D. $\boldsymbol{A}^\text{T}+\boldsymbol{A}$与$\boldsymbol{B}^\text{T}+\boldsymb >设 $n$ 阶方阵 $A$ 满足 $A^2 - 3A + 2E = O$ ，证明 $A$ 可相似对角化。 >[!note] 解析 ->$(A - 2E)(A - E) = 0$，容易得出 $A$ 的特征值只能是 $1$ 或者 $2$ 。 ->$\therefore \text{rank}(A - 2E) + \text{rank}(A - E) \leq n$ +>设 $A\boldsymbol x=\lambda \boldsymbol x$， +>$(A - 2E)(A - E) = 0\Rightarrow (\lambda^2-3\lambda+2)\boldsymbol x=0$，则 $A$ 的特征值只能是 $1$ 或者 $2$ 。 +>$\text{rank}(A - 2E) + \text{rank}(A - E) \leq n$ >$\text{rank}((A - E) - (A - 2E)) = n \leq \text{rank}(A - E)$ >$\therefore \text{rank}(A - 2E) + \text{rank}(A - E) = n$ >$\therefore \text{dim}N(A-2E)+\text{dim}N( A - E) = n$