Need Help with positive definite matrices

  • Thread starter Basil4000
  • Start date
  • #1
3
0

Homework Statement


If A is positive definite, show that ## A = C C^T ## where ## C ## has orthogonal columns.

The Attempt at a Solution



So, I've got the first part figured out. Because ## A ## is symmetric, an orthogonal matrix ## P ## exists such that ## P^TA P = D = diag(\lambda_1,...,\lambda_n) ## where ## \lambda_i > 0 ## because A is positive definite. Next I've defined ## B = diag(\sqrt{\lambda_1},...,\sqrt{\lambda_n}) ## Then I wrote ## C = P^TB P ## then ## C C^T = (P^T B P) (P^T B P)^T = (P^T B P) P^T B^T P = P^T B B P = P^T D P = A ##

So I'm at the last step and I'm stuck on how to show that C has orthogonal columns. Any hints?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,109
278
C has orthogonal columns if and only if CtC is a diagonal matrix (you should be able to check that this claim is true easily- the off diagonal entries of Ct C are the inner products of the columns of C).

Also PDPt = A, not PtDP.
 
  • #3
3
0
Right, that's just sloppyness on my part typing that out. I haven't learned anything about inner products yet. Is there a way to look at this differently?
 
  • #4
Office_Shredder
Staff Emeritus
Science Advisor
Gold Member
4,109
278
Inner product is the same as dot product. The typical definition of orthogonal vectors is that their dot product is zero, if you're working with a different definition you'll have to say what it is
 
  • #5
3
0
I just read your first reply again. I get it now. I was pretty tired last night. Thanks a lot!
 

Related Threads on Need Help with positive definite matrices

  • Last Post
Replies
14
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
2
Replies
31
Views
2K
Replies
15
Views
5K
Replies
19
Views
1K
  • Last Post
Replies
3
Views
1K
Replies
2
Views
1K
  • Last Post
Replies
7
Views
2K
  • Last Post
Replies
2
Views
784
Replies
6
Views
2K
Top