0
$\begingroup$

The actual question looks like-
Let $A\in M_n(\Bbb{C})$ such that $\text{Im}(A)\cap \text{Ker}(A)=\{\theta\}$, where $\text{Im}(A)=\{AX \mid X\in\Bbb{C^n}\}$ and $\text{Ker}(A)=\{X\in\Bbb{C}^n\mid AX=\theta$}, then prove that there exists non-singular matrices $P$ and $D$ of orders $n\times n$ and $\text{rank}(A)\times\text{rank}(A)$ respectively such that $$ A=P \begin{pmatrix} D & 0 \\ 0 & 0 \\ \end{pmatrix} P^{-1} $$ Can anybody suggest me a proper solution to that question?
Thanks for assistance in advance.

$\endgroup$
0
$\begingroup$

You can consider a basis of $\mathbb{C}^n$ using the decomposition $\mathbb{C}^n = Im(A) \oplus Ker(A)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.