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Suppose $A$, $B$ are $n ×n$ positive definite matrices and I be the $n ×n$ identity matrix. Then which of the following are positive definite?

  1. $A+B$
  2. $ABA $
  3. $A^2+I$
  4. $AB$

My thoughts: From the given condtion we have that the eigenvalues of $A$ and $B$ are positive and then I need to find the eigenvalues of given matrices. But how can I find them? Please help. Thanks.

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Are you assuming that $A$ and $B$ are symmetric? Some people include that in the definition of "positive definite", others don't. – joriki Dec 12 '12 at 17:51

I suppose you are talking about real symmetric matrices.

  1. Is $x^T(A+B)x>0$ for every nonzero vector $x$?
  2. Is $x^T(ABA)x = (Ax)^TB(Ax)>0$ for every nonzero vector $x$? (By the way, if $x\not=0$, is $Ax\not=0$?)
  3. Is $x^T(A+I)x>0$ for every nonzero vector $x$?
  4. Is $AB$ necessarily symmetric? That is, is $AB$ always equal to $(AB)^T=BA$?
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