Wednesday, 30 September 2015

linear algebra - Positive semidefinite versus all the eigenvalues having non-negative real parts





  1. Suppose matrix A with all its eigenvalues having non-negative real parts, can we get that xTAx0 holds for any vector x?


  2. Suppose matrix A is positive semidefinite, B is a positive definite diagonal matrix with the same dimension as A. Do all the eigenvalues of AB have nonnegative real parts?



Answer



For your first question, the answer is negative. A counter-examples is as follows.



Consider A=[110001] which has two non-negative real parts, but we have xTAx=98 when x=[11].


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