• Support PF! Buy your school textbooks, materials and every day products Here!

Symmetric Matrix and Definiteness

  • #1

Homework Statement


If A is a symmetric matrix, what can you say about the definiteness of A^2? Explain.



Homework Equations


I believe I need to use the face that A^2=SD^2S^-1.

I know that if all the eigenvalues of a symmetric matrix are positive then the matrix is positive definiteness and if the eigenvalues are positive and zero then the matrix is semidefinite.

Not sure where to go from here.
 

Answers and Replies

  • #2
22,097
3,278
So, we need to show that A2 is positive definite? What does this mean?
 
  • #3
I don't know if A^2 will be positive definite. Oh, wait..... if A^2= SD^2S^-1 then the diagonal matrix squared will only give positive or zero eigenvalues so A^2 will be positive semidefinite unless A is invertible then it would be positive definite. Do I have to worry about the orthogonal matrix and its inverse?
 
  • #4
22,097
3,278
Yes, A2 will always be positive definite, unless A is zero...
 

Related Threads for: Symmetric Matrix and Definiteness

  • Last Post
Replies
15
Views
3K
  • Last Post
Replies
2
Views
1K
Replies
4
Views
1K
Replies
23
Views
4K
  • Last Post
Replies
9
Views
7K
  • Last Post
Replies
3
Views
7K
  • Last Post
Replies
2
Views
708
  • Last Post
Replies
2
Views
1K
Top