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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.2012-12-12

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