问题

问题296: Consider A,B∈Rn×n . Suppose that A and B are similar. Show that there is an invertible matrix S such that Null(B)={x∈Rn∣∃y∈ Null(A),x=S−1y}


Question 3. Consider $A, B \in R ^{n \times n}$. Suppose that $A$ and $B$ are similar.

  1. Show that $A^2$ and $B^2$ are similar.
  2. Show that, for any integer $n \geq 3, A^n$ and $B^n$ are similar.
  3. Show that, if $A^2=I$, then $B^2=I$.
  4. Show that, if $A$ is invertible, then $B$ is invertible and $A^{-1}$ and $B^{-1}$ are similar.
  5. Show that there is an invertible matrix $S$ such that $\operatorname{Null}(B)=\left\{x \in R ^n \mid \exists y \in\right.$ $\left.\operatorname{Null}(A), x=S^{-1} y\right\}$.
  6. Show that $n(A)=n(B)$ and $\operatorname{rank}(A)=\operatorname{rank}(B)$.

待解决
提问于4月17日 · 阅读 75

解答
等待解答

添加微信可以直接提问(请注明数学答疑)

最后修改于4月17日

添加新讨论

提交新的问题
点此拍照题目

前一篇:问题295: 一个矩阵B的n次方把正方形区域变成了什么图

下一篇:问题297: an=∫x^n√1+x², 则lim n+1an+1/an=?

相关文章