
Linear Algebra Linear Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 17th, 2019, 12:57 PM  #1 
Newbie Joined: Dec 2018 From: Tel Aviv Posts: 4 Thanks: 0  Can somebody see whether my intuition re diagonalizability is correct?
Hi guys, can anybody see whether my intuition is correct? True or False? Let A and B be matrices of n x n. 1. If A and B are diagonalizable and they have the same characteristic polynomial, then A and B are similar. 2. If A and B are row equivalent and A is diagonalizable, then B is diagonalizable. My intuitive answer is "false" to 1, and "true" to 2. However, I am not sure, and either way, I would ideally like to be able to prove it... Many thanks! Last edited by skipjack; January 18th, 2019 at 04:12 AM. 
January 17th, 2019, 06:55 PM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 645 Thanks: 408 Math Focus: Dynamical systems, analytic function theory, numerics 
Hints: 1. If a matrix is diagonalizable it means its eigenvectors form a basis (show this). So both $A$ and $B$ have $n$ eigenvectors which you can order as $\{v_1,\dotsc,v_n\}$ and $\{w_1,\dotsc,w_n\}$ respectively. Let $C$ be the linear transformation which maps $v_i \mapsto w_i$ and note that $C$ must be invertible (prove this). This should be enough. 2. Row operations can (and should) be thought of as special linear transformations (sometimes called elementary operators). The important thing here is they are invertible. Now, $A$ is row equivalent to $B$ means you can factor as $A = E_kE_{k1}\dots E_1B$ where $E_i$'s are elementary. Now apply both sides of this to an eigenvector for $A$ and see what happens. 

Tags 
correct, diagonalizability, diagonalizable, intuition, matrix 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Please fix my intuition  dodo  Algebra  4  December 16th, 2018 11:14 AM 
Intuition behind differential element dA?  triplekite  Differential Equations  3  June 11th, 2013 04:22 PM 
Vector Intuition check  taylor_1989_2012  Algebra  1  March 14th, 2013 08:26 AM 
A better intuition?  Sefrez  Calculus  1  August 16th, 2011 03:17 PM 
Very simple, just need some intuition.  Sefrez  Calculus  1  February 24th, 2011 02:58 AM 