Eigenvalues of a positive definite matrix times a matrix with eigenvalues with positive real part

Question

Suppose that $A$ is a positive definite matrix and that $B$ is a matrix whose eigenvalues are complex but have strictly positive real parts. Can it be shown that the eigenvalues of $AB$ also have positive real parts? Is there an easy way to prove this?

Answer 1

For example: $$A = \pmatrix{1 & -1\cr -1 & 2\cr},\ B = \pmatrix{1 & 5\cr -1 & 1\cr},\ AB = \pmatrix{2 & 4\cr -3 & -3\cr}$$ Then $A$ is positive definite, $B$ has eigenvalues $1 \pm i \sqrt{5}$ with positive real part, but $AB$ has eigenvalues $-1/2 \pm i \sqrt{23}/2$ with negative real part.

Answer 2

No. Consider $B=\pmatrix{3&-5\\ 1&-1}$, whose eigenvalues (namely, $1\pm i$) have positive real parts. Let $A=\pmatrix{t&0\\ 0&1}$ with $t>0$. Since $AB\to\pmatrix{0&0\\ 1&-1}$ when $t\to0$, $AB$ has an eigenvalue with negative real part when $t$ is small.