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Symmetric positive definite matrix

  • #1
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Homework Statement


(i) Show that if [itex]A[/itex] is symmetric positive semi-definite, then there exists a symmetric matrix [itex]B[/itex] such that [itex]A=B^2[/itex].

(ii) Let [itex]A[/itex] be symmetric positive definite. Find a matrix [itex]B[/itex] such that [itex]A=B^2[/itex].


Homework Equations





The Attempt at a Solution



For part 1, I used:

[tex] B = Q\sqrt{\Lambda} Q^T[/tex]

So that,

[tex]B^T B = (Q\sqrt{\Lambda} Q^T)^T Q\sqrt{\Lambda} Q^T[/tex]
[tex]=Q\sqrt{\Lambda} Q^T Q\sqrt{\Lambda} Q^T[/tex]
[tex]=Q\Lambda Q^T[/tex]
[tex]=A[/tex]

I am assuming this cannot be used for part 2. Any help is much appreciated!
 

Answers and Replies

  • #2
MathematicalPhysicist
Gold Member
4,186
168
Why should it be different?

http://en.wikipedia.org/wiki/Positive-definite_matrix

semi positive means non-negative and positive is positive, this means that the eigenvalues are either way ≥0, so you can define a square root of a diagonal matrix in both cases as the square roots of the eigenvalues.
 
  • #3
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