wu :: forums
« wu :: forums - Have you seen this kind of matrix? »

Welcome, Guest. Please Login or Register.
Jun 8th, 2023, 11:46am

RIDDLES SITE WRITE MATH! Home Home Help Help Search Search Members Members Login Login Register Register
   wu :: forums
   riddles
   putnam exam (pure math)
(Moderators: Eigenray, Grimbal, SMQ, william wu, towr, Icarus)
   Have you seen this kind of matrix?
« Previous topic | Next topic »
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print
   Author  Topic: Have you seen this kind of matrix?  (Read 1272 times)
cuckoo
Junior Member
**





   


Gender: male
Posts: 57
Have you seen this kind of matrix?  
« on: Mar 27th, 2009, 8:28am »
Quote Quote Modify Modify

The matrix M is built from two vectors: a[1, m] and b[1, n]. M(i,j)=a(i)+b(j).
Do you know any property about this kind of matrices? 3x!
 Cheesy
IP Logged
towr
wu::riddles Moderator
Uberpuzzler
*****



Some people are average, some are just mean.

   


Gender: male
Posts: 13730
Re: Have you seen this kind of matrix?  
« Reply #1 on: Mar 27th, 2009, 11:29am »
Quote Quote Modify Modify

The matrix is the addition of two rank 1 matrices and so has rank 1 or 0. And I think that means it at most has one non-zero eigenvalue.
IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
Obob
Senior Riddler
****





   


Gender: male
Posts: 489
Re: Have you seen this kind of matrix?  
« Reply #2 on: Mar 27th, 2009, 2:10pm »
Quote Quote Modify Modify

The rank can be 2:  the vectors [0,1] & [0,1] give the matrix [[0,1],[1,2]], which is invertible.  But the rank can't be any bigger than 2.
IP Logged
cuckoo
Junior Member
**





   


Gender: male
Posts: 57
Re: Have you seen this kind of matrix?  
« Reply #3 on: Mar 27th, 2009, 8:24pm »
Quote Quote Modify Modify

you are right.
Thank you all, guys! Cheesy
 
on Mar 27th, 2009, 2:10pm, Obob wrote:
The rank can be 2:  the vectors [0,1] & [0,1] give the matrix [[0,1],[1,2]], which is invertible.  But the rank can't be any bigger than 2.  

IP Logged
cuckoo
Junior Member
**





   


Gender: male
Posts: 57
Re: Have you seen this kind of matrix?  
« Reply #4 on: Mar 27th, 2009, 8:31pm »
Quote Quote Modify Modify

conversely, if the rank of a matrix is less than or equal to 2, can it be represented in the form [a_i+b_j] ?
 
IP Logged
Obob
Senior Riddler
****





   


Gender: male
Posts: 489
Re: Have you seen this kind of matrix?  
« Reply #5 on: Mar 27th, 2009, 10:18pm »
Quote Quote Modify Modify

No; for instance, viewing it as a map R^n->R^m, the vector (1,1,...,1) is always in the image of both it and its transpose.  This is not a property of all matrices of rank at most 2.
IP Logged
Eigenray
wu::riddles Moderator
Uberpuzzler
*****






   


Gender: male
Posts: 1948
Re: Have you seen this kind of matrix?  
« Reply #6 on: Mar 27th, 2009, 11:33pm »
Quote Quote Modify Modify

If it's a square matrix, to compute the characteristic polynomial we need only find the coefficient of tn-2.  Thus
det( M - t I ) = (-t)n-2 [ t2 - S(a+b) t + S(a)S(b) - n <a,b> ],
where S(x) = xi.
« Last Edit: Mar 27th, 2009, 11:36pm by Eigenray » IP Logged
Pages: 1  Reply Reply Notify of replies Notify of replies Send Topic Send Topic Print Print

« Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board