From 99b5a6644c3521110b07a6fa846c8a7e15fa2d1a Mon Sep 17 00:00:00 2001 From: Cym10x Date: Mon, 12 Jan 2026 00:49:30 +0800 Subject: [PATCH] =?UTF-8?q?=E7=94=A8=E7=A7=A9=E7=9A=84=E4=B8=8D=E7=AD=89?= =?UTF-8?q?=E5=BC=8F=E2=80=9C=E5=A4=B9=E9=80=BC=E2=80=9D=E5=87=BA=E7=A1=AE?= =?UTF-8?q?=E5=88=87=E5=80=BC?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- 素材/用秩的不等式“夹逼”出确切值.md | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 素材/用秩的不等式“夹逼”出确切值.md diff --git a/素材/用秩的不等式“夹逼”出确切值.md b/素材/用秩的不等式“夹逼”出确切值.md new file mode 100644 index 0000000..60acc1f --- /dev/null +++ b/素材/用秩的不等式“夹逼”出确切值.md @@ -0,0 +1,9 @@ + +>[!information] 做题思路 +>通过矩阵的秩的不等式，最大限度限制所求的表达式的取值范围，或者将其**限制到一个具体的值**. +>在希望求一个矩阵的秩的确切值时，也可以考虑用不等式关系来“夹逼”，常见的不等式： +>1. $\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB})\le\min{\{\mathrm{rank}\boldsymbol{A}, \mathrm{rank}\boldsymbol{B}\}}$ +>2. $\mathrm{rank}(\boldsymbol{A+B})<\mathrm{rank}\boldsymbol A+\mathrm{rank}\boldsymbol B$ +>3. 矩阵加边不会减小秩； +> +>特别的，在遇到诸如 $AB=O$ 的情况，务必要想到$\mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}-n\le\mathrm{rank}(\boldsymbol{AB}) \Rightarrow \mathrm{rank}\boldsymbol{A}+\mathrm{rank}\boldsymbol{B}\le n$ \ No newline at end of file