It is currently Mon Oct 23, 2017 12:47 pm


All times are UTC [ DST ]




Post new topic Reply to topic  [ 2 posts ] 
Author Message
 Post subject: Invertible matrix
PostPosted: Sun Jan 24, 2016 10:40 pm 
Administrator
Administrator
User avatar

Joined: Sat Nov 07, 2015 6:12 pm
Posts: 803
Location: Larisa
Consider the matrices $A \in \mathcal{M}_{m \times n}$ and $B \in \mathcal{M}_{n \times m}$. If $AB +\mathbb{I}_m$ is invertible prove that $BA+\mathbb{I}_n$ is also invertible.

(Romania, 2012)

_________________
Imagination is much more important than knowledge.
Image


Top
Offline Profile  
Reply with quote  

 Post subject: Re: Invertible matrix
PostPosted: Fri Aug 26, 2016 5:07 pm 

Joined: Sat Nov 14, 2015 6:32 am
Posts: 127
Location: Melbourne, Australia
What the question basically asks is if $-1$ is a zero of the essentially same characteristic polynomials. $AB$ and $BA$ have quite similar characteristic polynomials. In fact if denote $p(x)$ the polynomial of $AB$, then the polynomial of $BA$ will be $q(x)= x^{n-m} p(x)$. It is easy to see that $-1$ cannot be an eigenvalue of the $AB$ matrix, otherwise it wouldn't be invertible. Now, let us assume that $BA$ is not invertible. Then it must have an eigenvalue of $-1$ and let $\mathbf{x}$ be the corresponding eigenvector. Hence:

$$\left ( BA \right )\mathbf{x}= -\mathbf{x} \Rightarrow AB \left ( A \mathbf{x} \right )= -A\mathbf{x}$$

meaning that $AB$ has an eigenvalue of $-1$ which is a contradiction. The result follows.

_________________
$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$


Top
Offline Profile  
Reply with quote  

Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 2 posts ] 

All times are UTC [ DST ]


Mathimatikoi Online

Users browsing this forum: No registered users and 1 guest


You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

Search for:
Jump to:  
cron
Powered by phpBB® Forum Software © phpBB Group Color scheme created with Colorize It.
Theme created StylerBB.net