Suppose matrix A with all its eigenvalues having non-negative real parts, can we get that xTAx≥0 holds for any vector x?
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=[1−10001] which has two non-negative real parts, but we have xTAx=−98 when x=[11].
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