June 21st, 2019, 12:59 AM  #1 
Newbie Joined: Jun 2019 From: London Posts: 9 Thanks: 0  eigenvalues
A permutation matrix of order n is a matrix of size n X n, composed of 0 and 1, that the sum (in the field of real numbers) of elements for each of its columns and each row is equal to 1. Let λ1, λ2, ..., λ5 be the proper numbers of the permutation of the order5. Find λ ∗ = min  λi . With Gaussian elimination, i found that λ = 1. However there must be 5 eigenvalues and there must be complex values. Hw are they being calculated?

June 21st, 2019, 03:38 AM  #2  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,226 Thanks: 908 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
$\displaystyle \lambda ^5  1 = ( \lambda  1)(\lambda ^4 + \lambda ^3 + \lambda ^2 + \lambda + 1) = 0$ We would have to solve the equation $\displaystyle \lambda ^4 + \lambda ^3 + \lambda ^2 + \lambda + 1 = 0$, which is a quartic equation. It looks ugly and it is ugly. There's a way to do it but I don't recommend it, myself. I've never been able to do one yet. So let's go to the geometry. In this case (the 5th roots of unity) the roots start at $\displaystyle \lambda _0 = 1$ and progress in a circle in the complex plane, at evenly spaced intervals over the unit circle. So we have 4 more roots: $\displaystyle \lambda _1 = e^{i (1 \cdot 2 \pi /5 )}$, $\displaystyle \lambda _2 = e^{i (2 \cdot 2\pi /5)}$, etc. Or more compactly: $\displaystyle \lambda _n = e^{i (n \cdot 2 \pi/ 5 )}$ where n = 1, 2, 3, 4. This can be a handy trick to have in your toolbelt. Dan  

Tags 
eigenvalues 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
eigenvalues and eigenvectors  matamat19  Linear Algebra  5  April 15th, 2019 03:51 PM 
Eigenvalues  Luiz  Linear Algebra  2  September 24th, 2015 05:32 PM 
Eigenvalues and Eigenvectors  bonildo  Linear Algebra  4  June 8th, 2012 07:02 PM 
eigenvalues  MacLaurin  Real Analysis  1  July 18th, 2009 07:44 PM 
eigenvalues  alpacino  Calculus  1  February 24th, 2009 02:10 PM 