Eigenvalues of product of p.d. Matrix with upper-triangular Matrix

Question

Let $A$ be a positive definite matrix (positive eigenvalues). Let $B$ be an upper triangular matrix, with ones in its main diagonal (i.e. all its eigenvalues are 1). Is there anything I can say about the eigenvalues of $AB$ ? I would like to find a way to prove that $AB$ has positive eigenvalues, if that's true. Thanks.

Answer 1

Consider the following example, where $AB$ has not even real eigenvalues. Consider the positive definite matrix: $$A = \begin{bmatrix} 2&1&2\\ 1&1&1\\ 2&1&3 \end{bmatrix} $$ with eigenvalues $\lambda_1 = 0.308$, $\lambda_2=0.6431,$ $\lambda_3=5.0489.$ Also, consider the upper triangular matrix: $$B=\begin{bmatrix} 1&1&1\\ 0&1&1\\ 0&0&1 \end{bmatrix}. $$

However, $$AB=\begin{bmatrix} 2&3&5\\ 1&2&3\\ 2&3&6 \end{bmatrix} $$

with eigenvalues $\lambda_1 = 9.3711,$ $\lambda_2=0.3144+0.0885i$, $\lambda_3=0.3144-0.0885i$.