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I have studied positive definite matrices . And I came across this exercise.

I can show that A+B is a positive definite matrix by the definition of positive definite matrix

Also in part (2) I can say AB is not possitive definite as it not necessarily symmetric

In part (3) I can conclude that A^2 is positive definite because all its eigenvalues are positive since A is positive definite

But I am not getting idea to proceed in rest of the parts

Please help.

  • 1
    $\begingroup$ If $A$ is symmetric, the last 2 are equal to $A^2$. $\endgroup$ – Paul Oct 11 '17 at 12:47
  • $\begingroup$ Oh yes. Thanks a lot. What about 4th and 5th part $\endgroup$ – Abhishek Chandra Oct 11 '17 at 12:50
  • $\begingroup$ Does (vii) read $A A^{\top}$? $\endgroup$ – Travis Oct 11 '17 at 12:56
  • $\begingroup$ @Travis yes it does $\endgroup$ – Abhishek Chandra Oct 11 '17 at 13:02

Hint Since $A$ is symmetric, (iv) and (v) coincide, as do (vi) and (vii).

For the former, for any vector $x \in \Bbb R^n$ (where $A, B$ are $n \times n$ matrices) we have $x^{\top} (A^T B A) x = (Ax)^{\top} B (Ax)$.

For the latter, as Paul pointed out in the comments, these quantities both coincide with $A^2$.

  • $\begingroup$ Can I just extend this question to positive semi definite ? Let's work that out $\endgroup$ – Abhishek Chandra Oct 11 '17 at 13:06
  • $\begingroup$ I think again only part 2 will not be positive semidefinite if a and b are positive semi definite . All rest will be positive semi definite also $\endgroup$ – Abhishek Chandra Oct 11 '17 at 13:09

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